reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem
  for N being strict normal Subgroup of G
  for K being characteristic Subgroup of N
  holds K is normal Subgroup of G
proof
  let N be strict normal Subgroup of G;
  let K be characteristic Subgroup of N;
  for a being Element of G holds K |^ a = the multMagma of K
  proof
    let a be Element of G;
    consider g being inner Automorphism of G such that
    A1: a is_inner_wrt g by Th32;
    Image(g|N) = N by Th33;
    then reconsider f = g|N as Automorphism of N by Th22;
    A2: Image(f|K) = the multMagma of K by Def3;
    for k being Element of G st k in K holds f.k = g.k by Th1,GROUP_2:40;
    then Image(g|K) = the multMagma of K & Image(g|K) = K |^ a
    by A1,A2,Th28,Th36;
    hence thesis;
  end;
  hence K is normal Subgroup of G by GROUP_3:def 13;
end;
