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(* 02110-1301 USA                                                     *)


(*****************************************************************************)
(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
(*****************************************************************************)
(*									     *)
(*	            The functor "- x a"                  		     *)
(*									     *)
(*****************************************************************************)
(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
(*****************************************************************************)

Require Export Binary_Products.
Require Export Functor.

Set Implicit Arguments.
Unset Strict Implicit.

(* De'finition du foncteur _xa : C -> C, pour tout a objet de C *)

Section fun_prod.

Variables (C : Category) (C1 : HasBinProd C) (a : C).

Definition Fun_prod_ob (c : C) := H_obj_prod C1 c a.

 Section fun_prod_map_def.

 Variable c1 c2 : C.

 Definition Fun_prod_mor (f : c1 --> c2) := Mor_prod C1 f (Id a).

 Lemma Fun_prod_map_law : Map_law Fun_prod_mor.
 Proof.
(* Goal: @Map_law (@Hom C c1 c2) (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a))) Fun_prod_mor *)
 unfold Map_law in |- *; intros f g H.
(* Goal: forall a : Ob C, @Equal (@Hom C (Fun_prod_ob a) (Fun_prod_ob a)) (@Fun_prod_mor a a (@Id C a)) (@Id C (Fun_prod_ob a)) *)
 unfold Fun_prod_mor, Mor_prod in |- *.
 apply
  (Map2_r (Op_together (C1 c2 a) _) (a1:=H_proj1_prod C1 c1 a o f)
     (a2:=H_proj1_prod C1 c1 a o g) (H_proj2_prod C1 c1 a o Id a)).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) c2) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) g) *)
 apply Comp_l; trivial.
 Qed.

 Canonical Structure Fun_prod_map :
   Map (c1 --> c2) (Fun_prod_ob c1 --> Fun_prod_ob c2) := Fun_prod_map_law.

 End fun_prod_map_def.

Lemma Fun_prod_comp_law : Fcomp_law Fun_prod_map.
Proof.
(* Goal: @Fcomp_law C C Fun_prod_ob Fun_prod_map *)
unfold Fcomp_law in |- *; simpl in |- *.
(* Goal: forall (a b c : Ob C) (f : Carrier (@Hom C a b)) (g : Carrier (@Hom C b c)), @Equal (@Hom C (Fun_prod_ob a) (Fun_prod_ob c)) (@Fun_prod_mor a c (@Comp C a b c f g)) (@Comp C (Fun_prod_ob a) (Fun_prod_ob b) (Fun_prod_ob c) (@Fun_prod_mor a b f) (@Fun_prod_mor b c g)) *)
intros c1 c2 c3 f g.
(* Goal: forall a : Ob C, @Equal (@Hom C (Fun_prod_ob a) (Fun_prod_ob a)) (@Fun_prod_mor a a (@Id C a)) (@Id C (Fun_prod_ob a)) *)
unfold Fun_prod_mor, Mor_prod in |- *.
(* *) apply
       Trans
        with
          (H_together C1
             (H_together C1 (H_proj1_prod C1 c1 a o f)
                (H_proj2_prod C1 c1 a o Id a) o H_proj1_prod C1 c2 a o g)
             (H_proj2_prod C1 c1 a o Id a)).
 apply
  (Map2_r (Op_together (C1 c3 a) _) (a1:=H_proj1_prod C1 c1 a o f o g)
     (a2:=H_together C1 (H_proj1_prod C1 c1 a o f)
            (H_proj2_prod C1 c1 a o Id a) o H_proj1_prod C1 c2 a o g)
     (H_proj2_prod C1 c1 a o Id a)).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) c3) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c3 (@H_proj1_prod C C1 c1 a) (@Comp C c1 c2 c3 f g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
(* *) apply Trans with ((H_proj1_prod C1 c1 a o f) o g).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) c3) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c3 (@H_proj1_prod C C1 c1 a) (@Comp C c1 c2 c3 f g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c2 c3 (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) g) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) c3) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c2 c3 (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) g) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Ass.
(* *) apply
       Trans
        with
          ((H_together C1 (H_proj1_prod C1 c1 a o f)
              (H_proj2_prod C1 c1 a o Id a) o H_proj1_prod C1 c2 a) o g).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) c3) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c2 c3 (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) g) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c2 c3 (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c2 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj1_prod C C1 c2 a)) g) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) c3) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c2 c3 (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c2 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj1_prod C C1 c2 a)) g) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Comp_r; apply Sym.
apply
 (Prf_eq1_prod (C1 c2 a) (H_proj1_prod C1 c1 a o f)
    (H_proj2_prod C1 c1 a o Id a)).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) (@Id C a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a))) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Ass1.
(* *) apply
       Trans
        with
          (H_together C1
             (H_together C1 (H_proj1_prod C1 c1 a o f)
                (H_proj2_prod C1 c1 a o Id a) o H_proj1_prod C1 c2 a o g)
             (H_together C1 (H_proj1_prod C1 c1 a o f)
                (H_proj2_prod C1 c1 a o Id a) o H_proj2_prod C1 c2 a o Id a)).
apply
 (Prf_map2_congl (Op_together (C1 c3 a) _) (b1:=H_proj2_prod C1 c1 a o Id a)
    (b2:=H_together C1 (H_proj1_prod C1 c1 a o f)
           (H_proj2_prod C1 c1 a o Id a) o H_proj2_prod C1 c2 a o Id a)
    (H_together C1 (H_proj1_prod C1 c1 a o f) (H_proj2_prod C1 c1 a o Id a)
     o H_proj1_prod C1 c2 a o g)).
(* *) apply
       Trans
        with
          ((H_together C1 (H_proj1_prod C1 c1 a o f)
              (H_proj2_prod C1 c1 a o Id a) o H_proj2_prod C1 c2 a) o 
           Id a).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@H_proj2_prod C C1 c1 a)) (@H_proj2_prod C C1 c2 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) (@Id C a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a))) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Comp_r.
(* *) apply
       Trans
        with
          (H_together C1 (H_proj1_prod C1 c1 a o f) (H_proj2_prod C1 c1 a)
           o H_proj2_prod C1 c2 a).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Sym.
apply
 (Prf_eq2_prod (C1 c2 a) (H_proj1_prod C1 c1 a o f) (H_proj2_prod C1 c1 a)).
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@H_proj2_prod C C1 c1 a)) (@H_proj2_prod C C1 c2 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) (@Id C a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a))) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Comp_r.
apply
 (Prf_map2_congl (Op_together (C1 c2 a) _) (b1:=H_proj2_prod C1 c1 a)
    (b2:=H_proj2_prod C1 c1 a o Id a) (H_proj1_prod C1 c1 a o f)).               
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@H_proj2_prod C C1 c1 a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a)) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) (@Id C a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a))) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Idr.
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) a) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_proj2_prod C C1 c2 a)) (@Id C a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a))) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Ass1.
(* Goal: @Equal (@Hom C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c3 a (C1 c3 a))) (@H_together C C1 c3 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) c3 (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) (@Obj_prod C c2 a (C1 c2 a)) a (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) (@Comp C (Fun_prod_ob c1) (Fun_prod_ob c2) (Fun_prod_ob c3) (@H_together C C1 c2 a (@Obj_prod C c1 a (C1 c1 a)) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) c1 c2 (@H_proj1_prod C C1 c1 a) f) (@Comp C (@Obj_prod C c1 a (C1 c1 a)) a a (@H_proj2_prod C C1 c1 a) (@Id C a))) (@H_together C C1 c3 a (@Obj_prod C c2 a (C1 c2 a)) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) c2 c3 (@H_proj1_prod C C1 c2 a) g) (@Comp C (@Obj_prod C c2 a (C1 c2 a)) a a (@H_proj2_prod C C1 c2 a) (@Id C a)))) *)
apply Sym.
apply
 (Eq3_prod (C1 c3 a) (H_proj1_prod C1 c2 a o g) (H_proj2_prod C1 c2 a o Id a)
    (H_together C1 (H_proj1_prod C1 c1 a o f) (H_proj2_prod C1 c1 a o Id a))).
Qed.

Lemma Fun_prod_id_law : Fid_law Fun_prod_map.
Proof.
(* Goal: @Fid_law C C Fun_prod_ob Fun_prod_map *)
unfold Fid_law in |- *; simpl in |- *.
(* Goal: forall a : Ob C, @Equal (@Hom C (Fun_prod_ob a) (Fun_prod_ob a)) (@Fun_prod_mor a a (@Id C a)) (@Id C (Fun_prod_ob a)) *)
unfold Fun_prod_mor, Mor_prod in |- *.
(* Goal: forall a0 : Ob C, @Equal (@Hom C (Fun_prod_ob a0) (Fun_prod_ob a0)) (@H_together C C1 a0 a (@Obj_prod C a0 a (C1 a0 a)) (@Comp C (@Obj_prod C a0 a (C1 a0 a)) a0 a0 (@H_proj1_prod C C1 a0 a) (@Id C a0)) (@Comp C (@Obj_prod C a0 a (C1 a0 a)) a a (@H_proj2_prod C C1 a0 a) (@Id C a))) (@Id C (Fun_prod_ob a0)) *)
intro c; unfold Fun_prod_ob in |- *.
(* *) apply
       Trans
        with
          (H_together C1 (Id (H_obj_prod C1 c a) o H_proj1_prod C1 c a)
             (H_proj2_prod C1 c a o Id a)).
apply
 (Prf_map2_congr (Op_together (C1 c a) _) (a1:=H_proj1_prod C1 c a o Id c)
    (a2:=Id (H_obj_prod C1 c a) o H_proj1_prod C1 c a)
    (H_proj2_prod C1 c a o Id a)).
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) c) (@Comp C (@Obj_prod C c a (C1 c a)) c c (@H_proj1_prod C C1 c a) (@Id C c)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) *)
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@Obj_prod C c a (C1 c a)) a a (@H_proj2_prod C C1 c a) (@Id C a))) (@Id C (@H_obj_prod C C1 c a)) *)
(* *) apply Trans with (H_proj1_prod C1 c a).
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@Comp C (@Obj_prod C c a (C1 c a)) a a (@H_proj2_prod C C1 c a) (@Id C a)) (@H_proj2_prod C C1 c a) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@H_proj2_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a)) *)
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a))) (@Id C (@H_obj_prod C C1 c a)) *)
apply Idr1.
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@H_proj2_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a)) *)
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a))) (@Id C (@H_obj_prod C C1 c a)) *)
apply Idl1.
(* *) apply
       Trans
        with
          (H_together C1 (Id (H_obj_prod C1 c a) o H_proj1_prod C1 c a)
             (Id (H_obj_prod C1 c a) o H_proj2_prod C1 c a)).
apply
 (Prf_map2_congl (Op_together (C1 c a) _) (b1:=H_proj2_prod C1 c a o Id a)
    (b2:=Id (H_obj_prod C1 c a) o H_proj2_prod C1 c a)
    (Id (H_obj_prod C1 c a) o H_proj1_prod C1 c a)).
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@Comp C (@Obj_prod C c a (C1 c a)) a a (@H_proj2_prod C C1 c a) (@Id C a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a)) *)
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a))) (@Id C (@H_obj_prod C C1 c a)) *)
(* *) apply Trans with (H_proj2_prod C1 c a).
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@Comp C (@Obj_prod C c a (C1 c a)) a a (@H_proj2_prod C C1 c a) (@Id C a)) (@H_proj2_prod C C1 c a) *)
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@H_proj2_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a)) *)
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a))) (@Id C (@H_obj_prod C C1 c a)) *)
apply Idr1.
(* Goal: @Equal (@Hom C (@Obj_prod C c a (C1 c a)) a) (@H_proj2_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a)) *)
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a))) (@Id C (@H_obj_prod C C1 c a)) *)
apply Idl1.
(* Goal: @Equal (@Hom C (@H_obj_prod C C1 c a) (@Obj_prod C c a (C1 c a))) (@H_together C C1 c a (@H_obj_prod C C1 c a) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) c (@Id C (@H_obj_prod C C1 c a)) (@H_proj1_prod C C1 c a)) (@Comp C (@H_obj_prod C C1 c a) (@H_obj_prod C C1 c a) a (@Id C (@H_obj_prod C C1 c a)) (@H_proj2_prod C C1 c a))) (@Id C (@H_obj_prod C C1 c a)) *)
apply (Prf_unique_together (Id (H_obj_prod C1 c a))).
Qed.

Canonical Structure Fun_prod :=
  Build_Functor Fun_prod_comp_law Fun_prod_id_law.

End fun_prod.

