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 &
  K./.(K,H)`*` is characteristic Subgroup of G./.H
  holds K is characteristic Subgroup of G
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;
  assume A2: K./.(K,H)`*` is characteristic Subgroup of G./.H;
  A3: (K,H)`*` = H by A1,GROUP_6:def 1;
  for phi being Automorphism of G
  for k being Element of G st k in K
  holds phi.k in K
  proof
    let phi be Automorphism of G;
    let k be Element of G;
    assume B1: k in K;
    Image(phi|H) = H by Def3;
    then consider sigma being Automorphism of G./.H such that
    B2: for x being Element of G holds sigma.(x*H) = (phi.x)*H
    by Th73;
    consider J being strict characteristic Subgroup of G./.H such that
    B3: J = (K./.(K,H)`*`) by A2;

    B4: for k1 being Element of G st k1*H in J holds k1 in K
    proof
      let k1 be Element of G;
      assume C1: k1*H in J;
      C2: k1*H = k1*(K,H)`*` by A1,GROUP_6:def 1;
      set x = k1*(K,H)`*`;
      consider a being Element of K such that
      C3: x = a*(K,H)`*` by B3,C1,C2,GROUP_2:def 15;

      reconsider a1 = a as Element of G by GROUP_2:42;
      C4: a1 in K;
      for j1 being object holds j1 in a*(K,H)`*` iff j1 in a1*H
      proof
        let j1 be object;
        thus j1 in a*(K,H)`*` implies j1 in a1*H
        proof
          assume j1 in a*(K,H)`*`;
          then consider g1 being Element of K such that
          D1: j1 = a*g1 & g1 in (K,H)`*` by GROUP_2:103;
          reconsider g=g1 as Element of G by GROUP_2:42;
          D2: j1 = a1*g by D1,GROUP_2:43;
          g in H by D1,A1,GROUP_6:def 1;
          hence j1 in a1*H by D2,GROUP_2:103;
        end;
        assume j1 in a1*H;
        then consider g1 being Element of G such that
        D1: j1 = a1*g1 & g1 in H by GROUP_2:103;
        reconsider g=g1 as Element of K by A1,D1,GROUP_2:42;
        D2: j1 = a*g by D1,GROUP_2:43;
        g in (K,H)`*` by D1,A1, GROUP_6:def 1;
        hence j1 in a*(K,H)`*` by D2,GROUP_2:103;
      end;
      then a1*H = x by TARSKI:2,C3
               .= k1*H by A1,GROUP_6:def 1;
      then (a1") * k1 in H by GROUP_2:114;
      then C5: (a1") * k1 in K by A1,GROUP_2:41;
      a1 * ((a1") * k1) = (a1 * a1") * k1 by GROUP_1:def 3
                       .= 1_G * k1 by GROUP_1:def 5
                       .= k1 by GROUP_1:def 4;
      hence k1 in K by C4,C5,GROUP_2:50;
    end;

    B5: for k1 being Element of G holds k1 in K iff k1*H in J
    proof
      let k1 be Element of G;
      thus k1 in K implies k1*H in J
      proof
        assume k1 in K;
        then reconsider k2=k1 as Element of K;
        C1: k2*((K,H)`*`) in J by B3, GROUP_2:def 15;
        for x being object holds x in k2*carr((K,H)`*`) iff x in k1*carr(H)
        proof
          let x be object;
          thus x in k2*carr((K,H)`*`) implies x in k1*carr(H)
          proof
            assume E1: x in k2*carr((K,H)`*`);
            x in k2*((K,H)`*`) iff
            ex g being Element of K st (x = k2*g & g in (K,H)`*`)
            by GROUP_2:103;
            then consider huh being Element of K such that
            E2: x = k2*huh & huh in (K,H)`*` by E1;
            E3: huh in H by A1,E2,GROUP_6:def 1;
            reconsider huh2=huh as Element of G by GROUP_2:42;
            set x2 = k1*huh2;
            x = k1*huh2 by E2,GROUP_2:43;
            hence thesis by E3,GROUP_2:27;
          end;
          assume x in k1*carr(H);
          then consider h1 being Element of G such that
          D1: x = k1*h1 & h1 in carr(H) by GROUP_2:27;
          reconsider h2=h1 as Element of K by A1,D1,GROUP_2:42;
          reconsider H1=H as normal Subgroup of K by A3;
          D2: the carrier of H = the carrier of ((K,H)`*`) by A1,GROUP_6:def 1;
          k2*h2 in k2*carr(H1) by D1,GROUP_2:27;
          hence x in k2*carr((K,H)`*`) by D1,D2,GROUP_2:43;
        end;
        then k2*carr(((K,H)`*`)) = k1*carr(H) by TARSKI:2
                                .= k1*H;
        hence k1*H in J by C1;
      end;
      thus k1*H in J implies k1 in K by B4;
    end;
    then k*H in J by B1;
    then reconsider kH = k*H as Element of G./.H by GROUP_2:42;
    sigma.(kH) in J by Th50,B1,B5;
    then sigma.(k*H) in J & sigma.(k*H) = (phi.k)*H by B2;
    hence phi.k in K by B4;
  end;
  hence K is characteristic Subgroup of G by Th50;
end;
