let x1, x2 be set ; 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; 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; ( 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
; ((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f))
now for x being set st x in {x1,x2} holds
(((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. xlet x be
set ;
( x in {x1,x2} implies (((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. x )assume A2:
x in {x1,x2}
;
(((x1,x2) --> (p1,p2)) * f) /. x = ((x1,x2) --> ((p1 (*) f),(p2 (*) f))) /. xthen
(
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;
verum end;
hence
((x1,x2) --> (p1,p2)) * f = (x1,x2) --> ((p1 (*) f),(p2 (*) f))
by Th1; verum