let C be Category; :: thesis:
let a, b, c be Object of C; :: thesis:
assume that
A1: Hom (c,a) <> {} and
A2: Hom (c,b) <> {} ; :: thesis:
let p1 be Morphism of c,a; :: thesis:
let p2 be Morphism of c,b; :: thesis:
thus ( c is_a_product_wrt p1,p2 implies for d being Object of C st Hom (d,a) <> {} & Hom (d,b) <> {} holds
( Hom (d,c) <> {} & ( for f being Morphism of d,a
for g being Morphism of d,b ex h being Morphism of d,c st
for k being Morphism of d,c holds
( ( p1 * k = f & p2 * k = g ) iff h = k ) ) ) ) :: thesis:

proof

assume that
dom p1 = c and
dom p2 = c and
A3: for d being Object of C
for f, g being Morphism of C st f in Hom (d,(cod p1)) & g in Hom (d,(cod p2)) holds
ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) ) ) ; :: according to CAT_3:def 15 :: thesis:
let d be Object of C; :: thesis:
assume that
A4: Hom (d,a) <> {} and
A5: Hom (d,b) <> {} ; :: thesis:
set f = the Morphism of d,a;
set g = the Morphism of d,b;
A6: cod p2 = b by A2, CAT_1:5;
then A7: the Morphism of d,b in Hom (d,(cod p2)) by A5, CAT_1:def 5;
A8: cod p1 = a by A1, CAT_1:5;
then the Morphism of d,a in Hom (d,(cod p1)) by A4, CAT_1:def 5;
then A9: ex h being Morphism of C st
( h in Hom (d,c) & ( for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = the Morphism of d,a & p2 (*) k = the Morphism of d,b ) iff h = k ) ) ) by A3, A7;
hence Hom (d,c) <> {} ; :: thesis:
let f be Morphism of d,a; :: thesis:
let g be Morphism of d,b; :: thesis:
A10: g in Hom (d,(cod p2)) by A5, A6, CAT_1:def 5;
f in Hom (d,(cod p1)) by A4, A8, CAT_1:def 5;
then consider h being Morphism of C such that
A11: h in Hom (d,c) and
A12: for k being Morphism of C st k in Hom (d,c) holds
( ( p1 (*) k = f & p2 (*) k = g ) iff h = k ) by A3, A10;
reconsider h = h as Morphism of d,c by A11, CAT_1:def 5;
take h ; :: thesis:
let k be Morphism of d,c; :: thesis:
A13: k in Hom (d,c) by A9, CAT_1:def 5;
( p1 * k = p1 (*) k & p2 * k = p2 (*) k ) by A1, A2, A9, CAT_1:def 13;
hence ( ( p1 * k = f & p2 * k = g ) iff h = k ) by A12, A13; :: thesis:

end;

assume A14: for d being Object of C st Hom (d,a) <> {} & Hom (d,b) <> {} holds
( Hom (d,c) <> {} & ( for f being Morphism of d,a
for g being Morphism of d,b ex h being Morphism of d,c st
for k being Morphism of d,c holds
( ( p1 * k = f & p2 * k = g ) iff h = k ) ) ) ; :: thesis:
thus ( dom p1 = c & dom p2 = c ) by A1, A2, CAT_1:5; :: according to CAT_3:def 15 :: thesis:
let d be Object of C; :: thesis:
let f, g be Morphism of C; :: thesis:
assume that
A15: f in Hom (d,(cod p1)) and
A16: g in Hom (d,(cod p2)) ; :: thesis:
A17: Hom (d,a) <> {} by A1, A15, CAT_1:5;
A18: cod p1 = a by A1, CAT_1:5;
then A19: f is Morphism of d,a by A15, CAT_1:def 5;
A20: cod p2 = b by A2, CAT_1:5;
then g is Morphism of d,b by A16, CAT_1:def 5;
then consider h being Morphism of d,c such that
A21: for k being Morphism of d,c holds
( ( p1 * k = f & p2 * k = g ) iff h = k ) by A14, A16, A20, A19, A17;
reconsider h9 = h as Morphism of C ;
take h9 ; :: thesis:
A22: Hom (d,c) <> {} by A14, A15, A16, A18, A20;
hence h9 in Hom (d,c) by CAT_1:def 5; :: thesis:
let k be Morphism of C; :: thesis:
assume k in Hom (d,c) ; :: thesis:
then reconsider k9 = k as Morphism of d,c by CAT_1:def 5;
( p1 (*) k = p1 * k9 & p2 (*) k = p2 * k9 ) by A1, A2, A22, CAT_1:def 13;
hence ( ( p1 (*) k = f & p2 (*) k = g ) iff h9 = k ) by A21; :: thesis: