reserve x,O for set,
  o for Element of O,
  G,H,I for GroupWithOperators of O,
  A, B for Subset of G,
  N for normal StableSubgroup of G,
  H1,H2,H3 for StableSubgroup of G,
  g1,g2 for Element of G,
  h1,h2 for Element of H1,
  h for Homomorphism of G,H;

theorem Th42:
  for G being strict GroupWithOperators of O holds G is trivial
  implies (1).G = G
proof
  let G be strict GroupWithOperators of O;
  reconsider H=G as StableSubgroup of G by Lm3;
  assume G is trivial;
  then ex x be object st the carrier of G = {x};
  then the carrier of H = {1_G} by TARSKI:def 1;
  hence thesis by Def8;
end;
