let x1, x2 be set ; :: thesis: for C being Category
for f, p1, p2 being Morphism of C st x1 <> x2 holds
((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f))
let C be Category; :: thesis: for f, p1, p2 being Morphism of C st x1 <> x2 holds
((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f))
let f, p1, p2 be Morphism of C; :: thesis: ( x1 <> x2 implies ((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f)) )
set F = (x1,x2) --> (p1,p2);
set F9 = (x1,x2) --> ((p1 (*) f),(p2 (*) f));
assume A1: x1 <> x2 ; :: thesis: ((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f))

now :: thesis: for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x

let x be set ; :: thesis: ( x in {x1,x2} implies (((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x )
assume A2: x in {x1,x2} ; :: thesis: (((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x
then ( x = x1 or x = x2 ) by TARSKI:def 2;
then ( ( ((x1,x2) --> (p1,p2)) /. x = p1 & ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x = p1 (*) f ) or ( ((x1,x2) --> (p1,p2)) /. x = p2 & ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x = p2 (*) f ) ) by A1, Th3;
hence (((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x by A2, Def5; :: thesis: verum

end;

hence ((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f)) by Th1; :: thesis: verum