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 Th33:
  for H being strict StableSubgroup of G holds (1).G "\/" H = H &
  H "\/" (1).G = H
proof
  let H be strict StableSubgroup of G;
  1_G in H by Lm17;
  then 1_G in carr H by STRUCT_0:def 5;
  then {1_G} c= carr H by ZFMISC_1:31;
  then
A1: {1_G} \/ carr H = carr H by XBOOLE_1:12;
  carr(1).G = {1_G} by Def8;
  hence thesis by A1,Th25;
end;
