let IT be Function of A,BOOLEAN; :: thesis: ( IT = p 'nand' q iff for x being Element of A holds IT . x = (p . x) 'nand' (q . x) )
A2: dom IT = A by FUNCT_2:def 1;

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

assume A3: IT = p 'nand' q ; :: thesis: for x being Element of A holds IT . x = (p . x) 'nand' (q . x)
let x be Element of A; :: thesis: IT . x = (p . x) 'nand' (q . x)
( dom p = A & dom q = A ) by FUNCT_2:def 1;
then dom (p 'nand' q) = A /\ A by Def1
.= A ;
hence IT . x = (p . x) 'nand' (q . x) by A3, Def1; :: thesis: verum

end;

A4: dom q = A by FUNCT_2:def 1;
A5: dom IT = A /\ A by FUNCT_2:def 1
.= (dom p) /\ (dom q) by A4, FUNCT_2:def 1 ;
assume for x being Element of A holds IT . x = (p . x) 'nand' (q . x) ; :: thesis: IT = p 'nand' q
then for x being set st x in dom IT holds
IT . x = (p . x) 'nand' (q . x) by A2;
hence IT = p 'nand' q by A5, Def1; :: thesis: verum