A in bool E by Def1;
then ( A /\ X c= A & A c= E ) by XBOOLE_1:17, ZFMISC_1:def 1;
then A /\ X c= E ;
then A /\ X in bool E by ZFMISC_1:def 1;
hence A /\ X is Subset of E by Def1; :: thesis: verum