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 H being strict Subgroup of G
  st (for K being strict Subgroup of G
      st card K = card H
      holds H = K)
  holds H is characteristic
proof
  let H be strict Subgroup of G;
  assume A1: for K being strict Subgroup of G st card K = card H holds H = K;
  H is characteristic
  proof
    let phi be Automorphism of G;
    Image(phi|H) = phi .: H by GRSOLV_1:def 3;
    then card H = card Image(phi|H) by Th19,GROUP_6:73;
    hence Image(phi|H) = the multMagma of H by A1;
  end;
  hence thesis;
end;
