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 Th11:
  for G1,G2,G3 being GroupWithOperators of O holds G1 is
StableSubgroup of G2 & G2 is StableSubgroup of G3 implies G1 is StableSubgroup
  of G3
proof
  let G1,G2,G3 be GroupWithOperators of O;
  assume that
A1: G1 is StableSubgroup of G2 and
A2: G2 is StableSubgroup of G3;
A3: G1 is Subgroup of G2 by A1,Def7;
A4: now
    let o be Element of O;
A5: the carrier of G1 c= the carrier of G2 by A3,GROUP_2:def 5;
    G1^o = (G2^o)|the carrier of G1 by A1,Def7
      .= ((G3^o)|the carrier of G2)|the carrier of G1 by A2,Def7
      .= (G3^o)|((the carrier of G2) /\ the carrier of G1) by RELAT_1:71;
    hence G1^o = (G3^o)|the carrier of G1 by A5,XBOOLE_1:28;
  end;
  G2 is Subgroup of G3 by A2,Def7;
  then G1 is Subgroup of G3 by A3,GROUP_2:56;
  hence thesis by A4,Def7;
end;
