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

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, Th7;
hence (((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 * f),(p2 * f))) /. x by A2, Def7; :: thesis: verum

end;

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