(***)
Opaque omit. intros G E A F H H0. elim H0. clear H0. intros
c H0. elim H0. clear H0. intros v H0. elim H0. clear H0. intros
vDIST H0. elim H0. clear H0. intros SURJ NOTDIST. induction c.
apply NOTDIST. unfold distinct. intuition.
(***)
apply NOTDIST. unfold distinct. intros i j i_not_j.
set(w:=comp_map_map F v). unfold not. intro wi_wj.
apply (IHc(omit v i)). apply(omit_preserves_distinct vDIST).
apply stillsurj2 with(j:=j). exact SURJ. exact i_not_j. exact
wi_wj. apply H. apply(omit_preserves_distinct vDIST). apply
stillsurj2 with(j:=j). exact SURJ. exact i_not_j. exact wi_wj.
Qed. Transparent omit.

(* Specifications: *)
(*************************************************************** I call the composition of a serial numbering of G with the left action map F the "F-derivative".  I also call the weird lengthy forall statement that is the second to last hypothesis of existbij2_c "P". existbij2_c says: if all injective serial numberings of G that F-cover A .m.u.s.t. have their F-derivative be nondistinct, then the existence of a single injective serial numbering of G that F-covers A and has a nondistinct F-derivative implies a contra- diction. But here is a single injective serial numbering of G that F-covers A.  (the "v" argument) Now suppose its F-derivative is distinct.  Then we have the desired product; we are immediately done. But if its F-derivative is nondistinct?  Well suppose P.  Then contradiction by existbij2_c, so P can't be true.  But if P is not true, then we have ~P which by forall-exists conversion, shows that exists injective serial numbering of G that F-covers A and its F-derivative is ~~distinct.  q.e.d.  (The forall- exists conversion is done by NNPP in the proof.  I think.) ***************************************************************)
Lemma existbij2:
forall(G:Setoid)(c:Nat)(v:seq c G)(V:full G=' seq_set v)
(V'':distinct v)
(E:Setoid)(A:part_set E)
(F:MAP G E)(sF:A=' seq_set(comp_map_map F v)),
exists k:Nat,exists w:seq k E,A=' seq_set w/\distinct w.