set A = (carr H1) /\ (carr H2);
1_ G in H2 by Th46;
then A1: 1_ G in the carrier of H2 ;
A2: now :: thesis: for g1, g2 being Element of G st g1 in (carr H1) /\ (carr H2) & g2 in (carr H1) /\ (carr H2) holds
g1 * g2 in (carr H1) /\ (carr H2)
let g1, g2 be Element of G; :: thesis: ( g1 in (carr H1) /\ (carr H2) & g2 in (carr H1) /\ (carr H2) implies g1 * g2 in (carr H1) /\ (carr H2) )
assume A3: ( g1 in (carr H1) /\ (carr H2) & g2 in (carr H1) /\ (carr H2) ) ; :: thesis: g1 * g2 in (carr H1) /\ (carr H2)
then ( g1 in carr H2 & g2 in carr H2 ) by XBOOLE_0:def 4;
then A4: g1 * g2 in carr H2 by Th74;
( g1 in carr H1 & g2 in carr H1 ) by ;
then g1 * g2 in carr H1 by Th74;
hence g1 * g2 in (carr H1) /\ (carr H2) by ; :: thesis: verum
end;
A5: now :: thesis: for g being Element of G st g in (carr H1) /\ (carr H2) holds
g " in (carr H1) /\ (carr H2)
let g be Element of G; :: thesis: ( g in (carr H1) /\ (carr H2) implies g " in (carr H1) /\ (carr H2) )
assume A6: g in (carr H1) /\ (carr H2) ; :: thesis: g " in (carr H1) /\ (carr H2)
then g in carr H2 by XBOOLE_0:def 4;
then A7: g " in carr H2 by Th75;
g in carr H1 by ;
then g " in carr H1 by Th75;
hence g " in (carr H1) /\ (carr H2) by ; :: thesis: verum
end;
1_ G in H1 by Th46;
then 1_ G in the carrier of H1 ;
then (carr H1) /\ (carr H2) <> {} by ;
hence ex b1 being strict Subgroup of G st the carrier of b1 = (carr H1) /\ (carr H2) by A2, A5, Th52; :: thesis: verum