proposition: Lemma Bolzano1 f x y: x <=r y -> (forall z, x <=r z -> z <=r y -> continuous_at f z) -> f x <=r \0r -> \0r <=r f y -> exists2 z, (x <=r z /\ z <=r y) & f z = \0r.
proof: Proof. move => xy /(Bolzano_hyp_simp xy) Hc la lb; apply: Bolzano => //. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)