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;
