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 K being strict characteristic Subgroup of G
  holds (Centralizer K) is characteristic Subgroup of G
proof
  let G be Group;
  let K be strict characteristic Subgroup of G;
  for phi being Automorphism of G
  for x being Element of G
  st x in Centralizer K
  holds phi.x in Centralizer K
  proof
    let phi be Automorphism of G;
    let x be Element of G;
    assume B1: x in Centralizer K;
    set y = phi.x;
    reconsider psi = phi" as Automorphism of G by GROUP_6:62;
    for k being Element of G st k in K holds y*k = k*y
    proof
      let k be Element of G;
      assume C1: k in K;
      set j = psi.k;
      phi.(x*j) = phi.(j*x) by B1,C1,Th50,Th60
               .= phi.j * phi.x by GROUP_6:def 6;
      then y * phi.(psi.k) = phi.(psi.k) * y by GROUP_6:def 6
                          .= k * y by Th4;
      hence y * k = k * y by Th4;
    end;
    then y is Element of Centralizer K by Th60;
    hence thesis;
  end;
  hence Centralizer K is characteristic Subgroup of G by Th50;
end;
