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 Th57:
  for G being strict GroupWithOperators of O holds G./.(Omega).G is trivial
proof
  let G be strict GroupWithOperators of O;
  reconsider G9=G as Group;
  reconsider H=the multMagma of (Omega).G as strict normal Subgroup of G by Lm6
;
A1: H = (Omega).G9;
  the carrier of G./.(Omega).G = Cosets H by Def14
    .= {the carrier of G} by A1,GROUP_2:142;
  hence thesis;
end;
