theorem Th58:
  (for g st g in H1 holds g in H2) implies H1 is Subgroup of H2
proof
  assume
A1: for g st g in H1 holds g in H2;
  the carrier of H1 c= the carrier of H2
  proof
    let x be object;
    assume x in the carrier of H1;
    then reconsider g = x as Element of H1;
    reconsider g as Element of G by Th42;
    g in H1;
    then g in H2 by A1;
    hence thesis;
  end;
  hence thesis by Th57;
end;
