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;

theorem Th17:
  ex psi being Automorphism of G
  st psi = phi" & Image(phi|Image(psi|H)) = the multMagma of H
proof
  reconsider psi = phi" as Automorphism of G by GROUP_6:62;
  take psi;
  thus psi = phi";
  for g being Element of G holds g in Image(phi|Image(psi|H)) iff g in H
  proof
    let g be Element of G;
    thus g in Image(phi|Image(psi|H)) implies g in H
    proof
      assume g in Image(phi|Image(psi|H));
      then consider a being Element of Image(psi|H) such that
      B1: g = (phi|Image(psi|H)).a by GROUP_6:45;
      M1: a in Image(psi|H) & a is Element of G by GROUP_2:42;
      then B2: phi.a = g by B1, Th1;
      consider b being Element of H such that
      B3: a = (psi|H).b
      by M1,GROUP_6:45;
      M2: b in H & b is Element of G by GROUP_2:42;
      then b = phi.(psi.b) by Th4
            .= g by M2,B2,B3,Th1;
      hence g in H;
    end;

    thus g in H implies g in Image(phi|Image(psi|H))
    proof
      assume B1: g in H;
      set a = (psi|H).g;
      B2: a in Image(psi|H)
      proof
        g in dom(psi|H) by B1,FUNCT_2:def 1;
        then (psi|H).g in (psi|H) .: (the carrier of H) by FUNCT_1:def 6;
        hence a in Image(psi|H) by GROUP_6:def 10;
      end;

      set K = Image(psi|H);
      set b = (phi|Image(psi|H)).a;

      B3: b in Image(phi|Image(psi|H))
      proof
        a in dom(phi|K) by B2, FUNCT_2:def 1;
        then (phi|K).a in (phi|K) .: (the carrier of K) by FUNCT_1:def 6;
        hence b in Image(phi|K) by GROUP_6:def 10;
      end;
      thus g in Image(phi|K)
      proof
        B4: psi.g = a by B1,Th1;
        a is Element of G by B2,GROUP_2:42;
        then (phi|K).a = phi.a by B2,Th1
                      .= g by B4,Th4;
        hence thesis by B3;
      end;
    end;
  end;
  hence Image(phi|Image(psi|H)) = the multMagma of H by GROUP_2:60;
end;
