thus ( X = Y implies for e being ext-real number holds
( e in X iff e in Y ) ) ; :: thesis: ( ( for e being ext-real number holds
( e in X iff e in Y ) ) implies X = Y )
assume for e being ext-real number holds
( e in X iff e in Y ) ; :: thesis: X = Y
then ( X c= Y & Y c= X ) by Def8;
hence X = Y by XBOOLE_0:def 10; :: thesis: verum