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

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

let x be set ; :: thesis: ( x in {x1,x2} implies (f * ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((f (*) p1),(f (*) p2))) /. x )
assume A2: x in {x1,x2} ; :: thesis: (f * ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((f (*) p1),(f (*) p2))) /. x
then ( x = x1 or x = x2 ) by TARSKI:def 2;
then ( ( ((x1,x2) --> (p1,p2)) /. x = p1 & ((x1,x2) --> ((f (*) p1),(f (*) p2))) /. x = f (*) p1 ) or ( ((x1,x2) --> (p1,p2)) /. x = p2 & ((x1,x2) --> ((f (*) p1),(f (*) p2))) /. x = f (*) p2 ) ) by A1, Th3;
hence (f * ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((f (*) p1),(f (*) p2))) /. x by A2, Def6; :: thesis: verum

end;

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