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;
reserve E for set,
  A for Action of O,E,
  C for Subset of G,
  N1 for normal StableSubgroup of H1;

theorem
  for E being non empty set, A being Action of O,E holds for X being
  Subset of E, a being Element of E st X is not empty holds a in
  the_stable_subset_generated_by(X,A) iff ex F being FinSequence of O, x being
  Element of X st Product(F,A).x = a by Lm30;
