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 Th39:
  for X,Y being StableSubgroup of H1, X9,Y9 being StableSubgroup
  of G st X = X9 & Y = Y9 holds X9 /\ Y9 = X /\ Y
proof
  let X,Y be StableSubgroup of H1;
  reconsider Z = X /\ Y as StableSubgroup of G by Th11;
  let X9,Y9 be StableSubgroup of G;
  assume
A1: X=X9 & Y=Y9;
  the carrier of X /\ Y = (carr X) /\ (carr Y) by Def25;
  then X9 /\ Y9 = Z by A1,Th18;
  hence thesis;
end;
