(* Proof of lem_equiv_comp_fun_sig                                         *)
intros.
exact
(lem_equiv_comp_sig (equiv_fun f (lem_fun_sig_is_fun f a s Afs)) a
(lem_sig_fun_equiv_is_rel f a s Afs)
(lem_sig_fun_equiv_is_equiv f a s Afs) (lem_sig_fun_equiv_dom f a s Afs)
s f Afs (lem_fun_equiv_prop f (lem_fun_sig_is_fun f a s Afs))).
Qed.
(* End of proof of lem_equiv_comp_fun_sig                                  *)

(***************************************************************************)
(* Comme on pouvait s'y attendre, f = equiv_comp_fun o projQuotient        *)
(***************************************************************************)
Lemma lem_equiv_comp_fun_prop :
forall (f a s : E) (Afs : fun_sig f a s),
f =
comp
(projQuotient (equiv_fun f (lem_fun_sig_is_fun f a s Afs)) a
(lem_sig_fun_equiv_is_rel f a s Afs)
(lem_sig_fun_equiv_is_equiv f a s Afs)
(lem_sig_fun_equiv_dom f a s Afs)) (equiv_comp_fun f a s Afs) a
(quotient (equiv_fun f (lem_fun_sig_is_fun f a s Afs)) a
(lem_sig_fun_equiv_is_rel f a s Afs)
(lem_sig_fun_equiv_is_equiv f a s Afs)
(lem_sig_fun_equiv_dom f a s Afs)) s
(lem_projQuotient_sig (equiv_fun f (lem_fun_sig_is_fun f a s Afs)) a
(lem_sig_fun_equiv_is_rel f a s Afs)
(lem_sig_fun_equiv_is_equiv f a s Afs)
(lem_sig_fun_equiv_dom f a s Afs)) (lem_equiv_comp_fun_sig f a s Afs).