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 Th40:
  N is StableSubgroup of H1 implies N is normal StableSubgroup of H1
proof
  assume N is StableSubgroup of H1;
  then reconsider N9 = N as StableSubgroup of H1;
  now
    reconsider N99=the multMagma of N as normal Subgroup of G by Lm6;
    let H be strict Subgroup of H1;
    assume
A1: H = the multMagma of N9;
    reconsider N as Subgroup of G by Def7;
    H1 is Subgroup of G & N99 is Subgroup of N by Def7,GROUP_2:57;
    hence H is normal by A1,GROUP_6:8;
  end;
  hence thesis by Def10;
end;
