let IT1, IT2 be Function of (DISJOINT_PAIRS A),the carrier of (NormForm A); :: thesis: ( ( for a being Element of DISJOINT_PAIRS A holds IT1 . a = {a} ) & ( for a being Element of DISJOINT_PAIRS A holds IT2 . a = {a} ) implies IT1 = IT2 )
assume A3: ( ( for a being Element of DISJOINT_PAIRS A holds IT1 . a = {a} ) & ( for a being Element of DISJOINT_PAIRS A holds IT2 . a = {a} ) ) ; :: thesis: IT1 = IT2

now

let a be Element of DISJOINT_PAIRS A; :: thesis: IT1 . a = IT2 . a
( IT1 . a = {a} & IT2 . a = {a} ) by A3;
hence IT1 . a = IT2 . a ; :: thesis: verum

end;

hence IT1 = IT2 by FUNCT_2:113; :: thesis: verum