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 characteristic Subgroup of G
  for K being characteristic Subgroup of N
  holds K is characteristic Subgroup of G
proof
  let N be characteristic Subgroup of G;
  let K be characteristic Subgroup of N;
  for g being Automorphism of G holds Image(g|K) = the multMagma of K
  proof
    let g be Automorphism of G;
    Image(g|N) = the multMagma of N by Def3;
    then reconsider f = g|N as Automorphism of N by Th22;
    A1: 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;
    hence Image(g|K) = the multMagma of K by A1,Th36;
  end;
  hence K is characteristic Subgroup of G by Def3;
end;
