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;
