intros r1 r2 r3 H H1 H2.
now apply H1, Rmult_eq_reg_r with r1.
Qed.

(* Specifications: *)
Theorem Rmult_min_distr_r :
forall r r1 r2 : R,
(0 <= r)%R ->
(Rmin r1 r2 * r)%R = Rmin (r1 * r) (r2 * r).
Theorem Rmult_min_distr_l :
forall r r1 r2 : R,
(0 <= r)%R ->
(r * Rmin r1 r2)%R = Rmin (r * r1) (r * r2).
Lemma Rmin_opp: forall x y, (Rmin (-x) (-y) = - Rmax x y)%R.
Lemma Rmax_opp: forall x y, (Rmax (-x) (-y) = - Rmin x y)%R.
Theorem exp_le :
forall x y : R,
(x <= y)%R -> (exp x <= exp y)%R.
Theorem Rinv_lt :
forall x y,
(0 < x)%R -> (x < y)%R -> (/y < /x)%R.
Theorem Rinv_le :
forall x y,
(0 < x)%R -> (x <= y)%R -> (/y <= /x)%R.
Theorem sqrt_ge_0 :
forall x : R,
(0 <= sqrt x)%R.
Lemma sqrt_neg : forall x, (x <= 0)%R -> (sqrt x = 0)%R.
Theorem Rabs_le :
forall x y,
(-y <= x <= y)%R -> (Rabs x <= y)%R.
Theorem Rabs_le_inv :
forall x y,
(Rabs x <= y)%R -> (-y <= x <= y)%R.
Theorem Rabs_ge :
forall x y,
(y <= -x \/ x <= y)%R -> (x <= Rabs y)%R.
Theorem Rabs_ge_inv :
forall x y,
(x <= Rabs y)%R -> (y <= -x \/ x <= y)%R.
Theorem Rabs_lt :
forall x y,
(-y < x < y)%R -> (Rabs x < y)%R.
Theorem Rabs_lt_inv :
forall x y,
(Rabs x < y)%R -> (-y < x < y)%R.
Theorem Rabs_gt :
forall x y,
(y < -x \/ x < y)%R -> (x < Rabs y)%R.
Theorem Rabs_gt_inv :
forall x y,
(x < Rabs y)%R -> (y < -x \/ x < y)%R.
Theorem Rmult_min_distr_r :
forall r r1 r2 : R,
(0 <= r)%R ->
(Rmin r1 r2 * r)%R = Rmin (r1 * r) (r2 * r).
Theorem Rmult_min_distr_l :
forall r r1 r2 : R,
(0 <= r)%R ->
(r * Rmin r1 r2)%R = Rmin (r * r1) (r * r2).
Lemma Rmin_opp: forall x y, (Rmin (-x) (-y) = - Rmax x y)%R.
Lemma Rmax_opp: forall x y, (Rmax (-x) (-y) = - Rmin x y)%R.
Theorem exp_le :
forall x y : R,
(x <= y)%R -> (exp x <= exp y)%R.
Theorem Rinv_lt :
forall x y,
(0 < x)%R -> (x < y)%R -> (/y < /x)%R.
Theorem Rinv_le :
forall x y,
(0 < x)%R -> (x <= y)%R -> (/y <= /x)%R.
Theorem sqrt_ge_0 :
forall x : R,
(0 <= sqrt x)%R.
Lemma sqrt_neg : forall x, (x <= 0)%R -> (sqrt x = 0)%R.
Theorem Rabs_le :
forall x y,
(-y <= x <= y)%R -> (Rabs x <= y)%R.
Theorem Rabs_le_inv :
forall x y,
(Rabs x <= y)%R -> (-y <= x <= y)%R.
Theorem Rabs_ge :
forall x y,
(y <= -x \/ x <= y)%R -> (x <= Rabs y)%R.
Theorem Rabs_ge_inv :
forall x y,
(x <= Rabs y)%R -> (y <= -x \/ x <= y)%R.
Theorem Rabs_lt :
forall x y,
(-y < x < y)%R -> (Rabs x < y)%R.
Theorem Rabs_lt_inv :
forall x y,
(Rabs x < y)%R -> (-y < x < y)%R.
Theorem Rabs_gt :
forall x y,
(y < -x \/ x < y)%R -> (x < Rabs y)%R.
Theorem Rabs_gt_inv :
forall x y,
(x < Rabs y)%R -> (y < -x \/ x < y)%R.