let x1, x2 be set ; :: thesis:
let C be Category; :: thesis:
let c be Object of C; :: thesis:
let p1, p2 be Morphism of C; :: thesis:
set F = (x1,x2) --> (p1,p2);
set I = {x1,x2};
assume A1: x1 <> x2 ; :: thesis:
thus ( c is_a_product_wrt p1,p2 implies c is_a_product_wrt (x1,x2) --> (p1,p2) ) :: thesis:

assume A2: c is_a_product_wrt p1,p2 ; :: thesis:
then ( dom p1 = c & dom p2 = c ) ;
hence (x1,x2) --> (p1,p2) is Projections_family of c,{x1,x2} by Th44; :: according to CAT_3:def 14 :: thesis:
let b be Object of C; :: thesis:
let F9 be Projections_family of b,{x1,x2}; :: thesis:
assume A3: cods ((x1,x2) --> (p1,p2)) = cods F9 ; :: thesis:
set f = F9 /. x1;
set g = F9 /. x2;
A4: x1 in {x1,x2} by TARSKI:def 2;
then (cods ((x1,x2) --> (p1,p2))) /. x1 = cod (((x1,x2) --> (p1,p2)) /. x1) by Def2;
then cod (F9 /. x1) = cod (((x1,x2) --> (p1,p2)) /. x1) by A3, A4, Def2;
then A5: cod (F9 /. x1) = cod p1 by A1, Th3;
A6: x2 in {x1,x2} by TARSKI:def 2;
then (cods ((x1,x2) --> (p1,p2))) /. x2 = cod (((x1,x2) --> (p1,p2)) /. x2) by Def2;
then cod (F9 /. x2) = cod (((x1,x2) --> (p1,p2)) /. x2) by A3, A6, Def2;
then A7: cod (F9 /. x2) = cod p2 by A1, Th3;
dom (F9 /. x2) = b by A6, Th41;
then A8: F9 /. x2 in Hom (b,(cod p2)) by A7;
dom (F9 /. x1) = b by A4, Th41;
then F9 /. x1 in Hom (b,(cod p1)) by A5;
then consider h being Morphism of C such that
A9: h in Hom (b,c) and
A10: for k being Morphism of C st k in Hom (b,c) holds
( ( p1 (*) k = F9 /. x1 & p2 (*) k = F9 /. x2 ) iff h = k ) by A2, A8;
take h ; :: thesis:
thus h in Hom (b,c) by A9; :: thesis:
let k be Morphism of C; :: thesis:
assume A11: k in Hom (b,c) ; :: thesis:
thus ( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) implies h = k ) :: thesis:

assume A12: for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ; :: thesis:
then (((x1,x2) --> (p1,p2)) /. x2) (*) k = F9 /. x2 by A6;
then A13: p2 (*) k = F9 /. x2 by A1, Th3;
(((x1,x2) --> (p1,p2)) /. x1) (*) k = F9 /. x1 by A4, A12;
then p1 (*) k = F9 /. x1 by A1, Th3;
hence h = k by A10, A11, A13; :: thesis:

end;

assume h = k ; :: thesis:
then A14: ( p1 (*) k = F9 /. x1 & p2 (*) k = F9 /. x2 ) by A10, A11;
let x be set ; :: thesis:
assume x in {x1,x2} ; :: thesis:
then ( x = x1 or x = x2 ) by TARSKI:def 2;
hence (((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x by A1, A14, Th3; :: thesis:

end;

assume A15: c is_a_product_wrt (x1,x2) --> (p1,p2) ; :: thesis:
then A16: (x1,x2) --> (p1,p2) is Projections_family of c,{x1,x2} ;
x2 in {x1,x2} by TARSKI:def 2;
then A17: dom (((x1,x2) --> (p1,p2)) /. x2) = c by A16, Th41;
x1 in {x1,x2} by TARSKI:def 2;
then dom (((x1,x2) --> (p1,p2)) /. x1) = c by A16, Th41;
hence ( dom p1 = c & dom p2 = c ) by A1, A17, Th3; :: according to CAT_3:def 15 :: thesis:
let d be Object of C; :: thesis:
let f, g be Morphism of C; :: thesis:
assume that
A18: f in Hom (d,(cod p1)) and
A19: g in Hom (d,(cod p2)) ; :: thesis:
( dom f = d & dom g = d ) by A18, A19, CAT_1:1;
then reconsider F9 = (x1,x2) --> (f,g) as Projections_family of d,{x1,x2} by Th44;
cods ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((cod p1),(cod p2)) by Th7
.= (x1,x2) --> ((cod f),(cod p2)) by A18, CAT_1:1
.= (x1,x2) --> ((cod f),(cod g)) by A19, CAT_1:1
.= cods F9 by Th7 ;
then consider h being Morphism of C such that
A20: h in Hom (d,c) and
A21: for k being Morphism of C st k in Hom (d,c) holds
( ( for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x ) iff h = k ) by A15;
take h ; :: thesis:
thus h in Hom (d,c) by A20; :: thesis:
let k be Morphism of C; :: thesis:
assume A22: k in Hom (d,c) ; :: thesis:
thus ( p1 (*) k = f & p2 (*) k = g implies h = k ) :: thesis:

assume A23: ( p1 (*) k = f & p2 (*) k = g ) ; :: thesis:

let x be set ; :: thesis:
assume x in {x1,x2} ; :: thesis:
then ( x = x1 or x = x2 ) by TARSKI:def 2;
then ( ( ((x1,x2) --> (p1,p2)) /. x = p1 & F9 /. x = f ) or ( ((x1,x2) --> (p1,p2)) /. x = p2 & F9 /. x = g ) ) by A1, Th3;
hence (((x1,x2) --> (p1,p2)) /. x) (*) k = F9 /. x by A23; :: thesis:

end;

end;