left; assumption.
elim H; intro.
right; assumption.
generalize (derivable_derive Rs 0 (Rs_derivable 0%R)); intro; elim H1;
intros l H2; rewrite H2 in H0.
generalize (derive_pt_eq_1 Rs 0 l (Rs_derivable 0%R) H2); intros;
cut (0 < - (l / 2))%R.
intro H4; elim (H3 (- (l / 2))%R H4).
intros; cut ((x / 2)%R <> 0%R).
intro H6; cut (Rabs (x / 2) < x)%R.
intro H7; generalize (H5 (x / 2)%R H6 H7); intro H8; rewrite Rs_0 in H8;
rewrite Rplus_0_l in H8; unfold Rminus in H8; rewrite Ropp_0 in H8;
rewrite Rplus_0_r in H8;
generalize (Rabs_def2 (Rs (x / 2) / (x / 2) + - l) (- (l / 2)) H8);
intro H9; elim H9; intros;
generalize
(Rplus_lt_compat_r l (Rs (x / 2) / (x / 2) + - l) (- (l / 2)) H10);
rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r;
replace (- (l / 2) + l)%R with (l / 2)%R.
intro H12; cut (l / 2 < 0)%R.
intro H13; generalize (Rlt_trans (Rs (x / 2) / (x / 2)) (l / 2) 0 H12 H13);
intro H14; unfold Rdiv in H14; cut (0 < x * / 2)%R.
intro H15;
generalize
(Rmult_lt_compat_r (x * / 2) (Rs (x * / 2) * / (x * / 2)) 0 H15 H14);
rewrite Rmult_0_l; rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_r; intro H16;
elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rs (x * / 2)) 0 (Rs_pos (x * / 2)) H16)).
apply prod_neq_R0.
generalize (cond_pos x); intro H16; red in |- *; intro H17;
rewrite H17 in H16; elim (Rlt_irrefl 0 H16).
apply Rinv_neq_0_compat; discrR.
apply Rmult_lt_0_compat;
[ apply (cond_pos x) | apply Rinv_0_lt_compat; prove_sup0 ].
assert (Hyp : (0 < 2)%R).
prove_sup0.
unfold Rdiv in |- *;
generalize (Rmult_lt_compat_r (/ 2) l 0 (Rinv_0_lt_compat 2 Hyp) H0);
rewrite Rmult_0_l; intro H13; exact H13.
pattern l at 3 in |- *; rewrite double_var.
ring.
cut (0 < x / 2)%R.
intro H7; apply Rabs_def1.
pattern x at 1 in |- *; replace (pos x) with (2 * (x / 2))%R.
pattern (x / 2)%R at 1 in |- *; replace (x / 2)%R with (x / 2 + 0)%R.
replace (2 * (x / 2))%R with (x / 2 + x / 2)%R.
apply Rplus_lt_compat_l; assumption.
ring.
apply Rplus_0_r.
pattern (pos x) at 2 in |- *; rewrite double_var.
ring.
apply Rlt_trans with 0%R.
rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply (cond_pos x).
assumption.
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos x) | apply Rinv_0_lt_compat; prove_sup0 ].
unfold Rdiv in |- *; apply prod_neq_R0.
generalize (cond_pos x); intro H6; red in |- *; intro H7; rewrite H7 in H6;
elim (Rlt_irrefl 0 H6).
apply Rinv_neq_0_compat; discrR.
assert (Hyp : (0 < 2)%R).
prove_sup0.
unfold Rdiv in |- *; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar;
generalize (Rmult_lt_compat_r (/ 2) l 0 (Rinv_0_lt_compat 2 Hyp) H0);
rewrite Rmult_0_l; intro H4; assumption.
Qed.

Lemma eq_sym : forall x y : R, x = y -> y = x.
intros; rewrite H; reflexivity.