Section uaxG_verif.
Variables (a' : A) (h : x --> G a').
Definition FT_i0 := SSC2_i (s x) h.
Definition FT_t := SSC2_t (s x) h.
(* construction de h# : Lim(EoQ) -> a' *)
Definition FT_h_diese : Lim_EoQ --> a' := FT_Apeps FT_i0 o FT_t.
(* unicite' de h# *)
Variable FT_h' : Lim_EoQ --> a'.
(* h = tau# o G(h#') *)
Hypothesis H : UA_eq FT_tau_d h FT_h'.
Definition FT_j0 := SSC2_i (s x) FT_tau_d.
Definition FT_t' := SSC2_t (s x) FT_tau_d.
(* pullback P pour t et (t' o h#') *)
Definition FT_P : Pullback1 FT_t (FT_t' o FT_h') :=
A_c (Fpulb FT_t (FT_t' o FT_h')).
Definition FT_b := Fibred1_prod FT_P.
Definition FT_p := Fibred1_p FT_P.
Definition FT_q := Fibred1_q FT_P.
(* P is preserved by G *)
Definition FT_GP' := G_c FT_P.
Definition FT_GP : Fpullback1 (Fpulb FT_t (FT_t' o FT_h') o_F G) :=
Build_Limit FT_GP'.
(* Le grand diagramme commute *)

Lemma FT_diag1 :
Pullback_eq1 (FMor G FT_t) (FMor G (FT_t' o FT_h'))
(FT_f FT_i0) (FT_f FT_j0).