Sqrt_reg.v

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(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

Require Import Rbase.
Require Import Rfunctions.
Require Import Ranalysis1.
Require Import R_sqrt.
Local Open Scope R_scope.

(**********)
Lemma sqrt_var_maj :
  forall h:R, Rabs h <= 1 -> Rabs (sqrt (1 + h) - 1) <= Rabs h.forall h : R,
Rabs h <= 1 -> Rabs (sqrt (1 + h) - 1) <= Rabs h
Proof.forall h : R,
Rabs h <= 1 -> Rabs (sqrt (1 + h) - 1) <= Rabs h
  intros; cut (0 <= 1 + h).h:R
H:Rabs h <= 1
0 <= 1 + h -> Rabs (sqrt (1 + h) - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  intro; apply Rle_trans with (Rabs (sqrt (Rsqr (1 + h)) - 1)).h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  destruct (total_order_T h 0) as [[Hlt|Heq]|Hgt].h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  repeat rewrite Rabs_left.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
- (sqrt (1 + h) - 1) <= - (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  unfold Rminus; do 2 rewrite <- (Rplus_comm (-1)).h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
- (-1 + sqrt (1 + h)) <= - (-1 + sqrt (1 + h)²)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  change (-1) with (-(1)).h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
- (- (1) + sqrt (1 + h)) <= - (- (1) + sqrt (1 + h)²)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive;
    apply Rplus_le_compat_l.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
- sqrt (1 + h) <= - sqrt (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Ropp_le_contravar; apply sqrt_le_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
(1 + h)² <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rle_0_sqr.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
(1 + h)² <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
(1 + h)² <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern (1 + h) at 2; rewrite <- Rmult_1_r; unfold Rsqr;
    apply Rmult_le_compat_l.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
1 + h <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
1 + h <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
    assumption.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rplus_lt_reg_l with 1; rewrite Rplus_0_r; rewrite Rplus_comm;
    unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
      rewrite Rplus_0_r.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h)² < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 2; rewrite <- sqrt_1; apply sqrt_lt_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
(1 + h)² < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rle_0_sqr.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
(1 + h)² < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  left; apply Rlt_0_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
(1 + h)² < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 2; rewrite <- Rsqr_1; apply Rsqr_incrst_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
1 + h < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
    assumption.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  left; apply Rlt_0_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) - 1 < 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rplus_lt_reg_l with 1; rewrite Rplus_0_r; rewrite Rplus_comm;
    unfold Rminus; rewrite Rplus_assoc; rewrite Rplus_opp_l;
      rewrite Rplus_0_r.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
sqrt (1 + h) < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 2; rewrite <- sqrt_1; apply sqrt_lt_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
1 + h < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
1 + h < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  left; apply Rlt_0_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hlt:h < 0
1 + h < 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 2; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
    assumption.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Heq:h = 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  rewrite Heq; rewrite Rplus_0_r; rewrite Rsqr_1; rewrite sqrt_1; right;
    reflexivity.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
Rabs (sqrt (1 + h) - 1) <= Rabs (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  repeat rewrite Rabs_right.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 <= sqrt (1 + h)² - 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  unfold Rminus; do 2 rewrite <- (Rplus_comm (-1));
    apply Rplus_le_compat_l.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) <= sqrt (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply sqrt_le_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 + h <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 + h <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rle_0_sqr.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 + h <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern (1 + h) at 1; rewrite <- Rmult_1_r; unfold Rsqr;
    apply Rmult_le_compat_l.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
    assumption.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h)² - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rle_ge; apply Rplus_le_reg_l with 1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 + 0 <= 1 + (sqrt (1 + h)² - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
    rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= sqrt (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 1; rewrite <- sqrt_1; apply sqrt_le_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  left; apply Rlt_0_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rle_0_sqr.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= (1 + h)²
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 1; rewrite <- Rsqr_1; apply Rsqr_incr_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left;
    assumption.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  left; apply Rlt_0_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
sqrt (1 + h) - 1 >= 0
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply Rle_ge; left; apply Rplus_lt_reg_l with 1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 + 0 < 1 + (sqrt (1 + h) - 1)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus;
    rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 < sqrt (1 + h)
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 1; rewrite <- sqrt_1; apply sqrt_lt_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 < 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  left; apply Rlt_0_1.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
0 <= 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 < 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Hgt:h > 0
1 < 1 + h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  pattern 1 at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
    assumption.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (sqrt (1 + h)² - 1) <= Rabs h
h:R
H:Rabs h <= 1
0 <= 1 + h
  rewrite sqrt_Rsqr.h:R
H:Rabs h <= 1
H0:0 <= 1 + h
Rabs (1 + h - 1) <= Rabs h
h:R
H:Rabs h <= 1
H0:0 <= 1 + h
0 <= 1 + h
h:R
H:Rabs h <= 1
0 <= 1 + h
  replace (1 + h - 1) with h; [ right; reflexivity | ring ].h:R
H:Rabs h <= 1
H0:0 <= 1 + h
0 <= 1 + h
h:R
H:Rabs h <= 1
0 <= 1 + h
  apply H0.h:R
H:Rabs h <= 1
0 <= 1 + h
  destruct (total_order_T h 0) as [[Hlt|Heq]|Hgt].h:R
H:Rabs h <= 1
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
Heq:h = 0
0 <= 1 + h
h:R
H:Rabs h <= 1
Hgt:h > 0
0 <= 1 + h
  rewrite (Rabs_left h Hlt) in H.h:R
H:- h <= 1
Hlt:h < 0
0 <= 1 + h
h:R
H:Rabs h <= 1
Heq:h = 0
0 <= 1 + h
h:R
H:Rabs h <= 1
Hgt:h > 0
0 <= 1 + h
  apply Rplus_le_reg_l with (- h).h:R
H:- h <= 1
Hlt:h < 0
- h + 0 <= - h + (1 + h)
h:R
H:Rabs h <= 1
Heq:h = 0
0 <= 1 + h
h:R
H:Rabs h <= 1
Hgt:h > 0
0 <= 1 + h
  rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc;
    rewrite Rplus_opp_r; rewrite Rplus_0_r; exact H.h:R
H:Rabs h <= 1
Heq:h = 0
0 <= 1 + h
h:R
H:Rabs h <= 1
Hgt:h > 0
0 <= 1 + h
  left; rewrite Heq; rewrite Rplus_0_r; apply Rlt_0_1.h:R
H:Rabs h <= 1
Hgt:h > 0
0 <= 1 + h
  left; apply Rplus_lt_0_compat.h:R
H:Rabs h <= 1
Hgt:h > 0
0 < 1
h:R
H:Rabs h <= 1
Hgt:h > 0
0 < h
  apply Rlt_0_1.h:R
H:Rabs h <= 1
Hgt:h > 0
0 < h
  apply Hgt.
Qed.

sqrt is continuous in 1

Lemma sqrt_continuity_pt_R1 : continuity_pt sqrt 1.continuity_pt sqrt 1
Proof.continuity_pt sqrt 1
  unfold continuity_pt; unfold continue_in;
    unfold limit1_in; unfold limit_in;
      unfold dist; simpl; unfold R_dist;
        intros.eps:R
H:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond 1 x /\ Rabs (x - 1) < alp ->
   Rabs (sqrt x - sqrt 1) < eps)
  set (alpha := Rmin eps 1).eps:R
H:eps > 0
alpha:=Rmin eps 1:R
exists alp : R,
  alp > 0 /\
  (forall x : R,
   D_x no_cond 1 x /\ Rabs (x - 1) < alp ->
   Rabs (sqrt x - sqrt 1) < eps)
  exists alpha; intros.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
alpha > 0 /\
(forall x : R,
 D_x no_cond 1 x /\ Rabs (x - 1) < alpha ->
 Rabs (sqrt x - sqrt 1) < eps)
  split.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
alpha > 0
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
forall x : R,
D_x no_cond 1 x /\ Rabs (x - 1) < alpha ->
Rabs (sqrt x - sqrt 1) < eps
  unfold alpha; unfold Rmin; case (Rle_dec eps 1); intro.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
r:eps <= 1
eps > 0
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
n:~ eps <= 1
1 > 0
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
forall x : R,
D_x no_cond 1 x /\ Rabs (x - 1) < alpha ->
Rabs (sqrt x - sqrt 1) < eps
  assumption.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
n:~ eps <= 1
1 > 0
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
forall x : R,
D_x no_cond 1 x /\ Rabs (x - 1) < alpha ->
Rabs (sqrt x - sqrt 1) < eps
  apply Rlt_0_1.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
forall x : R,
D_x no_cond 1 x /\ Rabs (x - 1) < alpha ->
Rabs (sqrt x - sqrt 1) < eps
  intros; elim H0; intros.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (sqrt x - sqrt 1) < eps
  rewrite sqrt_1; replace x with (1 + (x - 1)); [ idtac | ring ];
    apply Rle_lt_trans with (Rabs (x - 1)).eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (sqrt (1 + (x - 1)) - 1) <= Rabs (x - 1)
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) < eps
  apply sqrt_var_maj.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) <= 1
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) < eps
  apply Rle_trans with alpha.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) <= alpha
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
alpha <= 1
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) < eps
  left; apply H2.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
alpha <= 1
eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) < eps
  unfold alpha; apply Rmin_r.eps:R
H:eps > 0
alpha:=Rmin eps 1:R
x:R
H0:D_x no_cond 1 x /\ Rabs (x - 1) < alpha
H1:D_x no_cond 1 x
H2:Rabs (x - 1) < alpha
Rabs (x - 1) < eps
  apply Rlt_le_trans with alpha;
    [ apply H2 | unfold alpha; apply Rmin_l ].
Qed.

sqrt is continuous forall x>0

Lemma sqrt_continuity_pt : forall x:R, 0 < x -> continuity_pt sqrt x.forall x : R, 0 < x -> continuity_pt sqrt x
Proof.forall x : R, 0 < x -> continuity_pt sqrt x
  intros; generalize sqrt_continuity_pt_R1.x:R
H:0 < x
continuity_pt sqrt 1 -> continuity_pt sqrt x
  unfold continuity_pt; unfold continue_in;
    unfold limit1_in; unfold limit_in;
      unfold dist; simpl; unfold R_dist;
        intros.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sqrt x0 - sqrt x) < eps)
  cut (0 < eps / sqrt x).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  intro; elim (H0 _ H2); intros alp_1 H3.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  elim H3; intros.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  set (alpha := alp_1 * x).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  exists (Rmin alpha x); intros.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
Rmin alpha x > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x ->
 Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  split.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
Rmin alpha x > 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  change (0 < Rmin alpha x); unfold Rmin;
    case (Rle_dec alpha x); intro.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
r:alpha <= x
0 < alpha
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
n:~ alpha <= x
0 < x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold alpha; apply Rmult_lt_0_compat; assumption.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
n:~ alpha <= x
0 < x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x0 : R,
 D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
 Rabs (sqrt x0 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x0 : R,
D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp_1 ->
Rabs (sqrt x0 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  intros; replace x0 with (x + (x0 - x)); [ idtac | ring ];
    replace (sqrt (x + (x0 - x)) - sqrt x) with
    (sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1)).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
Rabs (sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1)) <
eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rabs_mult; rewrite (Rabs_right (sqrt x)).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rmult_lt_reg_l with (/ sqrt x).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 < / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
/ sqrt x *
(sqrt x * Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1)) <
/ sqrt x * eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rinv_0_lt_compat; apply sqrt_lt_R0; assumption.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
/ sqrt x *
(sqrt x * Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1)) <
/ sqrt x * eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
1 * Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
/ sqrt x * eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_1_l; rewrite Rmult_comm.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold Rdiv in H5.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  case (Req_dec x x0); intro.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x = x0
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite H7; unfold Rminus, Rdiv; rewrite Rplus_opp_r;
    rewrite Rmult_0_l; rewrite Rplus_0_r; rewrite Rplus_opp_r;
      rewrite Rabs_R0.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x = x0
0 < eps * / sqrt x0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rmult_lt_0_compat.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x = x0
0 < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x = x0
0 < / sqrt x0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  assumption.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x = x0
0 < / sqrt x0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rinv_0_lt_compat; rewrite <- H7; apply sqrt_lt_R0; assumption.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (sqrt (1 + (x0 - x) / x) - sqrt 1) <
eps * / sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply H5.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
D_x no_cond 1 (1 + (x0 - x) / x) /\
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  split.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
D_x no_cond 1 (1 + (x0 - x) / x)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold D_x, no_cond.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
True /\ 1 <> 1 + (x0 - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  split.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
True
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
1 <> 1 + (x0 - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  trivial.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
1 <> 1 + (x0 - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  cut ((x0 - x) * / x = 0).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0 -> False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  intro.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  elim (Rmult_integral _ _ H9); intro.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:x0 - x = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  elim H7.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:x0 - x = 0
x = x0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply (Rminus_diag_uniq_sym _ _ H10).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  assert (H11 := Rmult_eq_0_compat_r _ x H10).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
H11:/ x * x = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite <- Rinv_l_sym in H11.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
H11:1 = 0
False
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
H11:/ x * x = 0
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  elim R1_neq_R0; exact H11.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
H9:(x0 - x) * / x = 0
H10:/ x = 0
H11:/ x * x = 0
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro; rewrite H12 in H; elim (Rlt_irrefl _ H).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:1 = 1 + (x0 - x) / x
(x0 - x) * / x = 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  symmetry ; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r;
    unfold Rdiv in H8; exact H8.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
Rabs (1 + (x0 - x) / x - 1) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold Rminus; rewrite Rplus_comm; rewrite <- Rplus_assoc;
    rewrite Rplus_opp_l; rewrite Rplus_0_l; elim H6; intros.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rabs ((x0 + - x) / x) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold Rdiv; rewrite Rabs_mult.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rabs (x0 + - x) * Rabs (/ x) < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rabs_Rinv.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rabs (x0 + - x) * / Rabs x < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite (Rabs_right x).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rabs (x0 + - x) * / x < alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_comm; apply Rmult_lt_reg_l with x.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
0 < x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x * (/ x * Rabs (x0 + - x)) < x * alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x * (/ x * Rabs (x0 + - x)) < x * alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
1 * Rabs (x0 + - x) < x * alp_1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rabs (x0 + - x) < alpha
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rlt_le_trans with (Rmin alpha x).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rabs (x0 + - x) < Rmin alpha x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rmin alpha x <= alpha
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply H9.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
Rmin alpha x <= alpha
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rmin_l.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro; rewrite H10 in H; elim (Rlt_irrefl _ H).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rle_ge; left; apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps * / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:x <> x0
H8:D_x no_cond x x0
H9:Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro; rewrite H10 in H; elim (Rlt_irrefl _ H).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  assert (H7 := sqrt_lt_R0 x H).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:0 < sqrt x
sqrt x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x >= 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rle_ge; apply sqrt_positivity.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  left; apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1) =
sqrt (x + (x0 - x)) - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold Rminus; rewrite Rmult_plus_distr_l;
    rewrite Ropp_mult_distr_r_reverse; repeat rewrite <- sqrt_mult.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt (x * (1 + (x0 + - x) / x)) + - sqrt (x * 1) =
sqrt (x + (x0 + - x)) + - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r;
    unfold Rdiv; rewrite Rmult_comm; rewrite Rmult_assoc;
      rewrite <- Rinv_l_sym.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
sqrt (x + (x0 + - x) * 1) + - sqrt x =
sqrt (x + (x0 + - x)) + - sqrt x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_1_r; reflexivity.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro; rewrite H7 in H; elim (Rlt_irrefl _ H).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  left; apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  left; apply Rlt_0_1.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  left; apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  elim H6; intros.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  destruct (Rcase_abs (x0 - x)) as [Hlt|Hgt].x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite (Rabs_left (x0 - x) Hlt) in H8.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rplus_comm.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
0 <= (x0 + - x) / x + 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rplus_le_reg_l with (- ((x0 - x) / x)).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
- ((x0 - x) / x) + 0 <=
- ((x0 - x) / x) + ((x0 + - x) / x + 1)
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rplus_0_r; rewrite <- Rplus_assoc; rewrite Rplus_opp_l;
    rewrite Rplus_0_l; unfold Rdiv; rewrite <- Ropp_mult_distr_l_reverse.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
- (x0 - x) * / x <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rmult_le_reg_l with x.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
0 < x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
x * (- (x0 - x) * / x) <= x * 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
x * (- (x0 - x) * / x) <= x * 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc;
    rewrite <- Rinv_l_sym.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
- (x0 - x) * 1 <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  rewrite Rmult_1_r; left; apply Rlt_le_trans with (Rmin alpha x).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
- (x0 - x) < Rmin alpha x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
Rmin alpha x <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply H8.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
Rmin alpha x <= x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rmin_r.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:- (x0 - x) < Rmin alpha x
Hlt:x0 - x < 0
x <> 0
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  red; intro; rewrite H9 in H; elim (Rlt_irrefl _ H).x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1 + (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rplus_le_le_0_compat.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= 1
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  left; apply Rlt_0_1.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= (x0 + - x) / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold Rdiv; apply Rmult_le_pos.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= x0 + - x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  apply Rge_le; exact Hgt.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x1 : R,
   D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp ->
   Rabs (sqrt x1 - sqrt 1) < eps0)
eps:R
H1:eps > 0
H2:0 < eps / sqrt x
alp_1:R
H3:alp_1 > 0 /\
(forall x1 : R,
 D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
 Rabs (sqrt x1 - sqrt 1) < eps / sqrt x)
H4:alp_1 > 0
H5:forall x1 : R,
D_x no_cond 1 x1 /\ Rabs (x1 - 1) < alp_1 ->
Rabs (sqrt x1 - sqrt 1) < eps / sqrt x
alpha:=alp_1 * x:R
x0:R
H6:D_x no_cond x x0 /\ Rabs (x0 - x) < Rmin alpha x
H7:D_x no_cond x x0
H8:Rabs (x0 - x) < Rmin alpha x
Hgt:x0 - x >= 0
0 <= / x
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  left; apply Rinv_0_lt_compat; apply H.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps / sqrt x
  unfold Rdiv; apply Rmult_lt_0_compat.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < eps
x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < / sqrt x
  apply H1.x:R
H:0 < x
H0:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 1 x0 /\ Rabs (x0 - 1) < alp ->
   Rabs (sqrt x0 - sqrt 1) < eps0)
eps:R
H1:eps > 0
0 < / sqrt x
  apply Rinv_0_lt_compat; apply sqrt_lt_R0; apply H.
Qed.

sqrt is derivable for all x>0

Lemma derivable_pt_lim_sqrt :
  forall x:R, 0 < x -> derivable_pt_lim sqrt x (/ (2 * sqrt x)).forall x : R,
0 < x -> derivable_pt_lim sqrt x (/ (2 * sqrt x))
Proof.forall x : R,
0 < x -> derivable_pt_lim sqrt x (/ (2 * sqrt x))
  intros; set (g := fun h:R => sqrt x + sqrt (x + h)).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
derivable_pt_lim sqrt x (/ (2 * sqrt x))
  cut (continuity_pt g 0).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0 ->
derivable_pt_lim sqrt x (/ (2 * sqrt x))
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  intro; cut (g 0 <> 0).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0 -> derivable_pt_lim sqrt x (/ (2 * sqrt x))
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  intro; assert (H2 := continuity_pt_inv g 0 H0 H1).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:continuity_pt (/ g) 0
derivable_pt_lim sqrt x (/ (2 * sqrt x))
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold derivable_pt_lim; intros; unfold continuity_pt in H2;
    unfold continue_in in H2; unfold limit1_in in H2;
      unfold limit_in in H2; simpl in H2; unfold R_dist in H2.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((sqrt (x + h) - sqrt x) / h - / (2 * sqrt x)) <
  eps
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  elim (H2 eps H3); intros alpha H4.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((sqrt (x + h) - sqrt x) / h - / (2 * sqrt x)) <
  eps
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  elim H4; intros.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((sqrt (x + h) - sqrt x) / h - / (2 * sqrt x)) <
  eps
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  set (alpha1 := Rmin alpha x).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((sqrt (x + h) - sqrt x) / h - / (2 * sqrt x)) <
  eps
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  cut (0 < alpha1).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1 ->
exists delta : posreal,
  forall h : R,
  h <> 0 ->
  Rabs h < delta ->
  Rabs ((sqrt (x + h) - sqrt x) / h - / (2 * sqrt x)) <
  eps
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  intro; exists (mkposreal alpha1 H7); intros.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs ((sqrt (x + h) - sqrt x) / h - / (2 * sqrt x)) <
eps
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  replace ((sqrt (x + h) - sqrt x) / h) with (/ (sqrt x + sqrt (x + h))).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs (/ (sqrt x + sqrt (x + h)) - / (2 * sqrt x)) <
eps
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold inv_fct, g in H6; replace (2 * sqrt x) with (sqrt x + sqrt (x + 0)).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs
  (/ (sqrt x + sqrt (x + h)) -
   / (sqrt x + sqrt (x + 0))) < eps
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply H6.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
D_x no_cond 0 h /\ Rabs (h - 0) < alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  split.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
D_x no_cond 0 h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs (h - 0) < alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold D_x, no_cond.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
True /\ 0 <> h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs (h - 0) < alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  split.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
True
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <> h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs (h - 0) < alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  trivial.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <> h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs (h - 0) < alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply (not_eq_sym (A:=R)); exact H8.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs (h - 0) < alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r;
    apply Rlt_le_trans with alpha1.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Rabs h < alpha1
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
alpha1 <= alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  exact H9.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
alpha1 <= alpha
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold alpha1; apply Rmin_l.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs
  (/ (sqrt x + sqrt (x + x0)) -
   / (sqrt x + sqrt (x + 0))) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
sqrt x + sqrt (x + 0) = 2 * sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  rewrite Rplus_0_r; ring.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  cut (0 <= x + h).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h ->
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  intro; cut (0 < sqrt x + sqrt (x + h)).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h) ->
/ (sqrt x + sqrt (x + h)) =
(sqrt (x + h) - sqrt x) / h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  intro; apply Rmult_eq_reg_l with (sqrt x + sqrt (x + h)).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
(sqrt x + sqrt (x + h)) * / (sqrt x + sqrt (x + h)) =
(sqrt x + sqrt (x + h)) *
((sqrt (x + h) - sqrt x) / h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  rewrite <- Rinv_r_sym.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
1 =
(sqrt x + sqrt (x + h)) *
((sqrt (x + h) - sqrt x) / h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  rewrite Rplus_comm; unfold Rdiv; rewrite <- Rmult_assoc;
    rewrite Rsqr_plus_minus; repeat rewrite Rsqr_sqrt.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
1 = (x + h - x) * / h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  rewrite Rplus_comm; unfold Rminus; rewrite Rplus_assoc;
    rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
1 = 1
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
h <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  reflexivity.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
h <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply H8.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  left; apply H.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  assumption.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  red; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
H11:0 < sqrt x + sqrt (x + h)
sqrt x + sqrt (x + h) <> 0
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  red; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x + sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply Rplus_lt_le_0_compat.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 < sqrt x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 <= sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply sqrt_lt_R0; apply H.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
H10:0 <= x + h
0 <= sqrt (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply sqrt_positivity; apply H10.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  destruct (Rcase_abs h) as [Hlt|Hgt].x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hlt:h < 0
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  rewrite (Rabs_left h Hlt) in H9.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:- h < {| pos := alpha1; cond_pos := H7 |}
Hlt:h < 0
0 <= x + h
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply Rplus_le_reg_l with (- h).x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:- h < {| pos := alpha1; cond_pos := H7 |}
Hlt:h < 0
- h + 0 <= - h + (x + h)
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc;
    rewrite Rplus_opp_r; rewrite Rplus_0_r; left; apply Rlt_le_trans with alpha1.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:- h < {| pos := alpha1; cond_pos := H7 |}
Hlt:h < 0
- h < alpha1
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:- h < {| pos := alpha1; cond_pos := H7 |}
Hlt:h < 0
alpha1 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply H9.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:- h < {| pos := alpha1; cond_pos := H7 |}
Hlt:h < 0
alpha1 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold alpha1; apply Rmin_r.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x + h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply Rplus_le_le_0_compat.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= x
x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  left; assumption.x:R
H:0 < x
g:=fun h0 : R => sqrt x + sqrt (x + h0):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
H7:0 < alpha1
h:R
H8:h <> 0
H9:Rabs h < {| pos := alpha1; cond_pos := H7 |}
Hgt:h >= 0
0 <= h
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply Rge_le; apply Hgt.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
0 < alpha1
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold alpha1; unfold Rmin; case (Rle_dec alpha x); intro.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
r:alpha <= x
0 < alpha
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
n:~ alpha <= x
0 < x
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply H5.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
H1:g 0 <> 0
H2:forall eps0 : R,
eps0 > 0 ->
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alp ->
   Rabs ((/ g)%F x0 - (/ g)%F 0) < eps0)
eps:R
H3:0 < eps
alpha:R
H4:alpha > 0 /\
(forall x0 : R,
 D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
 Rabs ((/ g)%F x0 - (/ g)%F 0) < eps)
H5:alpha > 0
H6:forall x0 : R,
D_x no_cond 0 x0 /\ Rabs (x0 - 0) < alpha ->
Rabs ((/ g)%F x0 - (/ g)%F 0) < eps
alpha1:=Rmin alpha x:R
n:~ alpha <= x
0 < x
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply H.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
g 0 <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  unfold g; rewrite Rplus_0_r.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
sqrt x + sqrt x <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  cut (0 < sqrt x + sqrt x).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
0 < sqrt x + sqrt x -> sqrt x + sqrt x <> 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
0 < sqrt x + sqrt x
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  intro; red; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1).x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
H0:continuity_pt g 0
0 < sqrt x + sqrt x
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  apply Rplus_lt_0_compat; apply sqrt_lt_R0; apply H.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt g 0
  replace g with (fct_cte (sqrt x) + comp sqrt (fct_cte x + id))%F;
  [ idtac | reflexivity ].x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt
  (fct_cte (sqrt x) + comp sqrt (fct_cte x + id)) 0
  apply continuity_pt_plus.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt (fct_cte (sqrt x)) 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt (comp sqrt (fct_cte x + id)) 0
  apply continuity_pt_const; unfold constant, fct_cte; intro;
    reflexivity.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt (comp sqrt (fct_cte x + id)) 0
  apply continuity_pt_comp.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt (fct_cte x + id) 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt sqrt ((fct_cte x + id)%F 0)
  apply continuity_pt_plus.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt (fct_cte x) 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt id 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt sqrt ((fct_cte x + id)%F 0)
  apply continuity_pt_const; unfold constant, fct_cte; intro;
    reflexivity.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt id 0
x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt sqrt ((fct_cte x + id)%F 0)
  apply derivable_continuous_pt; apply derivable_pt_id.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
continuity_pt sqrt ((fct_cte x + id)%F 0)
  apply sqrt_continuity_pt.x:R
H:0 < x
g:=fun h : R => sqrt x + sqrt (x + h):R -> R
0 < (fct_cte x + id)%F 0
  unfold plus_fct, fct_cte, id; rewrite Rplus_0_r; apply H.
Qed.

(**********)
Lemma derivable_pt_sqrt : forall x:R, 0 < x -> derivable_pt sqrt x.forall x : R, 0 < x -> derivable_pt sqrt x
Proof.forall x : R, 0 < x -> derivable_pt sqrt x
  unfold derivable_pt; intros.x:R
H:0 < x
{l : R | derivable_pt_abs sqrt x l}
  exists (/ (2 * sqrt x)).x:R
H:0 < x
derivable_pt_abs sqrt x (/ (2 * sqrt x))
  apply derivable_pt_lim_sqrt; assumption.
Qed.

(**********)
Lemma derive_pt_sqrt :
  forall (x:R) (pr:0 < x),
    derive_pt sqrt x (derivable_pt_sqrt _ pr) = / (2 * sqrt x).forall (x : R) (pr : 0 < x),
derive_pt sqrt x (derivable_pt_sqrt x pr) =
/ (2 * sqrt x)
Proof.forall (x : R) (pr : 0 < x),
derive_pt sqrt x (derivable_pt_sqrt x pr) =
/ (2 * sqrt x)
  intros.x:R
pr:0 < x
derive_pt sqrt x (derivable_pt_sqrt x pr) =
/ (2 * sqrt x)
  apply derive_pt_eq_0.x:R
pr:0 < x
derivable_pt_lim sqrt x (/ (2 * sqrt x))
  apply derivable_pt_lim_sqrt; assumption.
Qed.

We show that sqrt is continuous for all x>=0

Remark : by definition of sqrt (as extension of Rsqrt on |R), we could also show that sqrt is continuous for all x

Lemma continuity_pt_sqrt : forall x:R, 0 <= x -> continuity_pt sqrt x.forall x : R, 0 <= x -> continuity_pt sqrt x
Proof.forall x : R, 0 <= x -> continuity_pt sqrt x
  intros; case (Rtotal_order 0 x); intro.x:R
H:0 <= x
H0:0 < x
continuity_pt sqrt x
x:R
H:0 <= x
H0:0 = x \/ 0 > x
continuity_pt sqrt x
  apply (sqrt_continuity_pt x H0).x:R
H:0 <= x
H0:0 = x \/ 0 > x
continuity_pt sqrt x
  elim H0; intro.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
continuity_pt sqrt x
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  unfold continuity_pt; unfold continue_in;
    unfold limit1_in; unfold limit_in;
      simpl; unfold R_dist; intros.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
exists alp : R,
  alp > 0 /\
  (forall x0 : R,
   D_x no_cond x x0 /\ Rabs (x0 - x) < alp ->
   Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  exists (Rsqr eps); intros.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
eps² > 0 /\
(forall x0 : R,
 D_x no_cond x x0 /\ Rabs (x0 - x) < eps² ->
 Rabs (sqrt x0 - sqrt x) < eps)
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  split.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
eps² > 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < eps² ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  change (0 < Rsqr eps); apply Rsqr_pos_lt.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
eps <> 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < eps² ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  red; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2).x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
forall x0 : R,
D_x no_cond x x0 /\ Rabs (x0 - x) < eps² ->
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  intros; elim H3; intros.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs (x0 - x) < eps²
Rabs (sqrt x0 - sqrt x) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  rewrite <- H1; rewrite sqrt_0; unfold Rminus; rewrite Ropp_0;
    rewrite Rplus_0_r; rewrite <- H1 in H5; unfold Rminus in H5;
      rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Rabs (sqrt x0) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  destruct (Rcase_abs x0) as [Hlt|Hgt] eqn:Heqs.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hlt:x0 < 0
Heqs:Rcase_abs x0 = left Hlt
Rabs (sqrt x0) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
Rabs (sqrt x0) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  unfold sqrt.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hlt:x0 < 0
Heqs:Rcase_abs x0 = left Hlt
Rabs
  match Rcase_abs x0 with
  | left _ => 0
  | right a =>
      Rsqrt_def.Rsqrt
        {|
        nonneg := x0;
        cond_nonneg := Rge_le x0 0 a |}
  end < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
Rabs (sqrt x0) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x rewrite Heqs.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hlt:x0 < 0
Heqs:Rcase_abs x0 = left Hlt
Rabs 0 < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
Rabs (sqrt x0) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  rewrite Rabs_R0; apply H2.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
Rabs (sqrt x0) < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  rewrite Rabs_right.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 < eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  apply Rsqr_incrst_0.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
(sqrt x0)² < eps²
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= sqrt x0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  rewrite Rsqr_sqrt.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
x0 < eps²
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= x0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= sqrt x0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  rewrite (Rabs_right x0 Hgt) in H5; apply H5.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= x0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= sqrt x0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  apply Rge_le; exact Hgt.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= sqrt x0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  apply sqrt_positivity; apply Rge_le; exact Hgt.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
0 <= eps
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  left; exact H2.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 = x
eps:R
H2:eps > 0
x0:R
H3:D_x no_cond x x0 /\ Rabs (x0 - x) < eps²
H4:D_x no_cond x x0
H5:Rabs x0 < eps²
Hgt:x0 >= 0
Heqs:Rcase_abs x0 = right Hgt
sqrt x0 >= 0
x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  apply Rle_ge; apply sqrt_positivity; apply Rge_le; exact Hgt.x:R
H:0 <= x
H0:0 = x \/ 0 > x
H1:0 > x
continuity_pt sqrt x
  elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H1 H)).
Qed.