case H; intros.
- red; left.
apply Rlt_mult_inv_pos; auto with real.
- rewrite <- H1.
red; right; ring.
Qed.

(* Specifications: *)
Lemma sqrt_div_alt :
forall x y : R, 0 < y -> sqrt (x / y) = sqrt x / sqrt y.
Lemma sqrt_div :
forall x y:R, 0 <= x -> 0 < y -> sqrt (x / y) = sqrt x / sqrt y.
Lemma sqrt_lt_0_alt :
forall x y : R, sqrt x < sqrt y -> x < y.
Lemma sqrt_lt_0 : forall x y:R, 0 <= x -> 0 <= y -> sqrt x < sqrt y -> x < y.
Lemma sqrt_lt_1_alt :
forall x y : R, 0 <= x < y -> sqrt x < sqrt y.
Lemma sqrt_lt_1 : forall x y:R, 0 <= x -> 0 <= y -> x < y -> sqrt x < sqrt y.
Lemma sqrt_le_0 :
forall x y:R, 0 <= x -> 0 <= y -> sqrt x <= sqrt y -> x <= y.
Lemma sqrt_le_1_alt :
forall x y : R, x <= y -> sqrt x <= sqrt y.
Lemma sqrt_le_1 :
forall x y:R, 0 <= x -> 0 <= y -> x <= y -> sqrt x <= sqrt y.
Lemma sqrt_neg_0 x : x <= 0 -> sqrt x = 0.
Lemma sqrt_inj : forall x y:R, 0 <= x -> 0 <= y -> sqrt x = sqrt y -> x = y.
Lemma sqrt_less_alt :
forall x : R, 1 < x -> sqrt x < x.
Lemma sqrt_less : forall x:R, 0 <= x -> 1 < x -> sqrt x < x.
Lemma sqrt_more : forall x:R, 0 < x -> x < 1 -> x < sqrt x.
Lemma sqrt_inv x : sqrt (/ x) = / sqrt x.
Lemma inv_sqrt_depr x : 0 < x -> / sqrt x = sqrt (/ x).
#[deprecated(since="8.16",note="Use sqrt_inv.")]
Notation inv_sqrt := inv_sqrt_depr.
Lemma sqrt_cauchy :
forall a b c d:R,
a * c + b * d <= sqrt (Rsqr a + Rsqr b) * sqrt (Rsqr c + Rsqr d).