let IT be Function of A,BOOLEAN; :: thesis: ( IT = p 'xor' q iff for x being Element of A holds IT . x = (p . x) 'xor' (q . x) )
A7: dom IT = A by FUNCT_2:def 1;

hereby :: thesis: ( ( for x being Element of A holds IT . x = (p . x) 'xor' (q . x) ) implies IT = p 'xor' q )

assume A8: IT = p 'xor' q ; :: thesis: for x being Element of A holds IT . x = (p . x) 'xor' (q . x)
let x be Element of A; :: thesis: IT . x = (p . x) 'xor' (q . x)
( dom p = A & dom q = A ) by FUNCT_2:def 1;
then dom (p 'xor' q) = A /\ A by Def3
.= A ;
hence IT . x = (p . x) 'xor' (q . x) by A8, Def3; :: thesis: verum

end;

A9: dom q = A by FUNCT_2:def 1;
A10: dom IT = A /\ A by FUNCT_2:def 1
.= (dom p) /\ (dom q) by A9, FUNCT_2:def 1 ;
assume for x being Element of A holds IT . x = (p . x) 'xor' (q . x) ; :: thesis: IT = p 'xor' q
then for x being set st x in dom IT holds
IT . x = (p . x) 'xor' (q . x) by A7;
hence IT = p 'xor' q by A10, Def3; :: thesis: verum