Definition fincl A B (AsubB : A `<=` B) (a : A) : B :=
[` (fsubsetP AsubB) _ (valP a)].
Definition fsub B A : {set B} := [set x : B | val x \in A].

Lemma fproperEcard A B :
(A `<` B) = (A `<=` B) && (#|` A| < #|` B|)%N.