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 G being finite Group
  for H being strict characteristic Subgroup of G
  for K being strict Subgroup of G
  st H is Subgroup of K
  holds H is normal Subgroup of K
proof
  let G be finite Group;
  let H be strict characteristic Subgroup of G;
  let K be strict Subgroup of G;
  assume A1: H is Subgroup of K;
  A2: for g being Element of G
  holds g in Ker (nat_hom H) iff g in H by GROUP_6:43;

  reconsider R = Ker ((nat_hom H)|K) as strict Subgroup of K;

  A3: for k being Element of K
  holds k in H iff k in Ker ((nat_hom H)|K)
  proof
    let k be Element of K;
    reconsider g=k as Element of G by GROUP_2:42;
    B1: g in K;
    thus k in H implies k in Ker ((nat_hom H)|K)
    proof
      assume C1: k in H;
      C2: g in K;
      (nat_hom H).g = 1_(G./.H) by A2,C1,GROUP_6:41;
      then ((nat_hom H)|K).g = 1_(G./.H) by C2,Th1;
      hence k in Ker ((nat_hom H)|K) by GROUP_6:41;
    end;
    thus k in Ker ((nat_hom H)|K) implies k in H
    proof
      assume C1: k in Ker ((nat_hom H)|K);
      ((nat_hom H)|K).g = (nat_hom H).g by B1,Th1;
      then (nat_hom H).g = 1_(G./.H) by C1,GROUP_6:41;
      then g in Ker (nat_hom H) by GROUP_6:41;
      hence k in H by GROUP_6:43;
    end;
  end;
  reconsider H1=H as strict Subgroup of K by A1;
  the multMagma of R = the multMagma of H1 by A3,GROUP_2:60;
  hence thesis;
end;
