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;
  set A = commutators(H,K);
  reconsider A as non empty Subset of G by GROUP_5:53;
  for phi being Automorphism of G holds phi .: A = A by Th53;
  hence [.H,K.] is characteristic Subgroup of G by Th45;
end;
