let s1, s2 be Function; :: thesis: ( dom s1 = (dom p) /\ (dom q) & ( for x being set st x in dom s1 holds
s1 . x = (p . x) 'xor' (q . x) ) & dom s2 = (dom p) /\ (dom q) & ( for x being set st x in dom s2 holds
s2 . x = (p . x) 'xor' (q . x) ) implies s1 = s2 )
assume that
A9: dom s1 = (dom p) /\ (dom q) and
A10: for x being set st x in dom s1 holds
s1 . x = (p . x) 'xor' (q . x) and
A11: dom s2 = (dom p) /\ (dom q) and
A12: for x being set st x in dom s2 holds
s2 . x = (p . x) 'xor' (q . x) ; :: thesis: s1 = s2
for x being object st x in dom s1 holds
s1 . x = s2 . x

proof

let x be object ; :: thesis: ( x in dom s1 implies s1 . x = s2 . x )
assume A13: x in dom s1 ; :: thesis: s1 . x = s2 . x
then s1 . x = (p . x) 'xor' (q . x) by A10;
hence s1 . x = s2 . x by A9, A11, A12, A13; :: thesis: verum

end;

hence s1 = s2 by A9, A11, FUNCT_1:2; :: thesis: verum