thus ( X = Y implies for n being natural number holds
( n in X iff n in Y ) ) ; :: thesis: ( ( for n being natural number holds
( n in X iff n in Y ) ) implies X = Y )
assume for n being natural number holds
( n in X iff n in Y ) ; :: thesis: X = Y
then ( X c= Y & Y c= X ) by Def12;
hence X = Y by XBOOLE_0:def 10; :: thesis: verum