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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*		                                 			     *)
(*	     The Special Adjoint Functor Theorem (Freyd's Theorem)	     *)
(*                ( The suficient condition : Second Proof )                 *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export Th_Adjoint.
Require Export Pres_Limits.
Require Export Comma_Complete.
Require Export Th_Initial.
Require Export Comma_UA.
Require Export FAFT_SSC2.
Set Implicit Arguments.
Unset Strict Implicit.
(****************************************)
(*  CONDITION SUFFISANTE                *)
(* 2nd METHODE                          *)
(****************************************)
Section freyd_th_2'.
Variables (A X : Category) (G : Functor A X) (A_c : Complete A)
(G_c : Continuous G) (s : forall x : X, SSC2 G x).
Section uaxG'.
Variable x : X.
Section ssc2_to_ssc1.
Let I := SSC2_I (s x).
Let f (i : I) := SSC2_f i.
Let a (i : I) := SSC2_a i.
Let k (i : I) := Build_Com_ob (f i).
Section ssc2_to_cond1.
Variable axh : Comma G x.
Definition SSC2_1_i := SSC2_i (s x) (Mor_com_ob axh).
Definition SSC2_1_t := SSC2_t (s x) (Mor_com_ob axh).

Lemma SSC2_1_f_com_law : Com_law (axf:=k SSC2_1_i) (bxg:=axh) SSC2_1_t.
Canonical Structure SSC2_1_f := Build_Com_arrow SSC2_1_f_com_law.