theorem
  H1 is Subgroup of H2 implies H1 /\ H3 is Subgroup of H2 /\ H3
proof
  assume H1 is Subgroup of H2;
  then (the carrier of H1) /\ (the carrier of H3) c= (the carrier of H2) /\ (
  the carrier of H3) by DefA5,XBOOLE_1:26;
  then
  the carrier of H1 /\ H3 c= (the carrier of H2) /\ (the carrier of H3) by Th80
;
  then the carrier of H1 /\ H3 c= the carrier of H2 /\ H3 by Th80;
  hence thesis by Th57;
end;
