theorem Th61:
  for H2 being strict Subgroup of G holds H1 is Subgroup of H2 iff
  H1 "\/" H2 = H2
proof
  let H2 be strict Subgroup of G;
  thus H1 is Subgroup of H2 implies H1 "\/" H2 = H2
  proof
    assume H1 is Subgroup of H2;
    then the carrier of H1 c= the carrier of H2 by GROUP_2:def 5;
    hence H1 "\/" H2 = gr carr H2 by XBOOLE_1:12
      .= H2 by Th31;
  end;
  thus thesis by Th60;
end;
