let x1, x2 be set ; for C being Category st x1 <> x2 holds
for p1, p2 being Morphism of (C opp) holds opp ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((opp p1),(opp p2))
let C be Category; ( x1 <> x2 implies for p1, p2 being Morphism of (C opp) holds opp ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((opp p1),(opp p2)) )
assume A1:
x1 <> x2
; for p1, p2 being Morphism of (C opp) holds opp ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((opp p1),(opp p2))
let p1, p2 be Morphism of (C opp); opp ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((opp p1),(opp p2))
set F = (x1,x2) --> (p1,p2);
set F9 = (x1,x2) --> ((opp p1),(opp p2));
now for x being set st x in {x1,x2} holds
(opp ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((opp p1),(opp p2))) /. xlet x be
set ;
( x in {x1,x2} implies (opp ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((opp p1),(opp p2))) /. x )assume A2:
x in {x1,x2}
;
(opp ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((opp p1),(opp p2))) /. xthen
(
x = x1 or
x = x2 )
by TARSKI:def 2;
then
( (
((x1,x2) --> (p1,p2)) /. x = p1 &
((x1,x2) --> ((opp p1),(opp p2))) /. x = opp p1 ) or (
((x1,x2) --> (p1,p2)) /. x = p2 &
((x1,x2) --> ((opp p1),(opp p2))) /. x = opp p2 ) )
by A1, Th3;
hence
(opp ((x1,x2) --> (p1,p2))) /. x = ((x1,x2) --> ((opp p1),(opp p2))) /. x
by A2, Def4;
verum end;
hence
opp ((x1,x2) --> (p1,p2)) = (x1,x2) --> ((opp p1),(opp p2))
by Th1; verum