thus ( X = Y implies for i being integer number holds
( i in X iff i in Y ) ) ; :: thesis: ( ( for i being integer number holds
( i in X iff i in Y ) ) implies X = Y )
assume for i being integer number holds
( i in X iff i in Y ) ; :: thesis: X = Y
then ( X c= Y & Y c= X ) by Def11;
hence X = Y by XBOOLE_0:def 10; :: thesis: verum