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 x being Element of G st x in H /\ K
  holds phi.x in H /\ K
  proof
    let phi be Automorphism of G;
    let x be Element of G;
    assume x in H /\ K;
    then B1: x in H & x in K by GROUP_2:82;
    then B2: phi.x in H by Th50;
    phi.x in K by B1,Th50;
    hence phi.x in H /\ K by B2, GROUP_2:82;
  end;
  hence H /\ K is characteristic Subgroup of G by Th50;
end;
