set A = (carr H1) /\ (carr H2);
1_ G in H2 by Lm17;
then A1: 1_ G in the carrier of H2 by STRUCT_0:def 5;
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 that
A3: g1 in (carr H1) /\ (carr H2) and
A4: g2 in (carr H1) /\ (carr H2) ; :: thesis: g1 * g2 in (carr H1) /\ (carr H2)
g2 in carr H2 by A4, XBOOLE_0:def 4;
then A5: g2 in H2 by STRUCT_0:def 5;
g1 in carr H2 by A3, XBOOLE_0:def 4;
then g1 in H2 by STRUCT_0:def 5;
then g1 * g2 in H2 by A5, Lm18;
then A6: g1 * g2 in carr H2 by STRUCT_0:def 5;
g2 in carr H1 by A4, XBOOLE_0:def 4;
then A7: g2 in H1 by STRUCT_0:def 5;
g1 in carr H1 by A3, XBOOLE_0:def 4;
then g1 in H1 by STRUCT_0:def 5;
then g1 * g2 in H1 by A7, Lm18;
then g1 * g2 in carr H1 by STRUCT_0:def 5;
hence g1 * g2 in (carr H1) /\ (carr H2) by A6, XBOOLE_0:def 4; :: thesis: verum
end;
A8: now :: thesis: for o being Element of O
for a being Element of G st a in (carr H1) /\ (carr H2) holds
(G ^ o) . a in (carr H1) /\ (carr H2)
let o be Element of O; :: thesis: for a being Element of G st a in (carr H1) /\ (carr H2) holds
(G ^ o) . a in (carr H1) /\ (carr H2)
let a be Element of G; :: thesis: ( a in (carr H1) /\ (carr H2) implies (G ^ o) . a in (carr H1) /\ (carr H2) )
assume A9: a in (carr H1) /\ (carr H2) ; :: thesis: (G ^ o) . a in (carr H1) /\ (carr H2)
then a in carr H2 by XBOOLE_0:def 4;
then a in H2 by STRUCT_0:def 5;
then (G ^ o) . a in H2 by Lm9;
then A10: (G ^ o) . a in carr H2 by STRUCT_0:def 5;
a in carr H1 by A9, XBOOLE_0:def 4;
then a in H1 by STRUCT_0:def 5;
then (G ^ o) . a in H1 by Lm9;
then (G ^ o) . a in carr H1 by STRUCT_0:def 5;
hence (G ^ o) . a in (carr H1) /\ (carr H2) by A10, XBOOLE_0:def 4; :: thesis: verum
end;
A11: 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 A12: g in (carr H1) /\ (carr H2) ; :: thesis: g " in (carr H1) /\ (carr H2)
then g in carr H2 by XBOOLE_0:def 4;
then g in H2 by STRUCT_0:def 5;
then g " in H2 by Lm19;
then A13: g " in carr H2 by STRUCT_0:def 5;
g in carr H1 by A12, XBOOLE_0:def 4;
then g in H1 by STRUCT_0:def 5;
then g " in H1 by Lm19;
then g " in carr H1 by STRUCT_0:def 5;
hence g " in (carr H1) /\ (carr H2) by A13, XBOOLE_0:def 4; :: thesis: verum
end;
1_ G in H1 by Lm17;
then 1_ G in the carrier of H1 by STRUCT_0:def 5;
then (carr H1) /\ (carr H2) <> {} by A1, XBOOLE_0:def 4;
hence ex b1 being strict StableSubgroup of G st the carrier of b1 = (carr H1) /\ (carr H2) by A2, A11, A8, Lm14; :: thesis: verum