proposition: Lemma continuous_inj_image_segmentP a b f : a <= b -> {within `[a, b], continuous f} -> {in `[a, b] &, injective f} -> forall y, reflect (exists2 x, x \in `[a, b] & f x = y) (y \in f @`[a, b]).
proof: Proof. move=> /continuous_inj_image_segment/[apply]/[apply]/predeqP + y => /(_ y) faby. by apply/(equivP idP); symmetry. Qed.
imports: mathcomp(all_ssreflect ssralg ssrint ssrnum finmap) mathcomp(matrix interval zmodp vector fieldext falgebra) mathcomp(mathcomp_extra boolp classical_sets functions) mathcomp(cardinality) ereal reals signed topology prodnormedzmodule normedtype derive sequences real_interval HB(structures)