let IT be Element of Funcs A,BOOLEAN ; :: thesis: ( IT = p 'xor' q iff for x being Element of A holds IT . x = (p . x) 'xor' (q . x) )

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

assume A11: 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)
A12: dom p = A by FUNCT_2:def 1;
dom q = A by FUNCT_2:def 1;
then dom (p 'xor' q) = A /\ A by A12, Def6
.= A ;
hence IT . x = (p . x) 'xor' (q . x) by A11, Def6; :: thesis: verum

end;

assume A13: for x being Element of A holds IT . x = (p . x) 'xor' (q . x) ; :: thesis: IT = p 'xor' q
A14: dom IT = A by FUNCT_2:def 1;
A15: dom q = A by FUNCT_2:def 1;
A16: dom IT = A /\ A by FUNCT_2:def 1
.= (dom p) /\ (dom q) by A15, FUNCT_2:def 1 ;
for x being set st x in dom IT holds
IT . x = (p . x) 'xor' (q . x) by A13, A14;
hence IT = p 'xor' q by A16, Def6; :: thesis: verum