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 Group
  for H,K being strict characteristic Subgroup of G
  holds H "\/" K is characteristic Subgroup of G
proof
  let G be Group;
  let H,K be strict characteristic Subgroup of G;
  for phi being Automorphism of G
  for g being Element of G st g in H "\/" K
  holds phi.g in H "\/" K
  proof
    let phi be Automorphism of G;
    let g be Element of G;
    assume g in H "\/" K;
    then g in H*K by GROUP_4:53;
    then consider h, k being Element of G such that
    B1: g = h*k and
    B2: h in carr H and
    B3: k in carr K;
    h in H by B2;
    then B4: phi.h in H by Th50;
    k in K by B3;
    then phi.k in K by Th50;
    then phi.h * phi.k in carr H*carr K by B4;
    then phi.g in H*K by B1,GROUP_6:def 6;
    hence phi.g in H "\/" K by GROUP_4:53;
  end;
  hence H "\/" K is characteristic Subgroup of G by Th50;
end;
