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 A being Subset of G
  holds A is Subset of Centralizer (Centralizer A)
proof
  let G be Group;
  let A be Subset of G;
  set H = Centralizer A;
  for g being object
  st g in A
  holds g in the carrier of Centralizer H
  proof
    let g be object;
    assume B1: g in A;
    then reconsider g as Element of G;
    for h being Element of G st h in H
    holds g*h = h*g by B1,Th57;
    then g is Element of Centralizer H by Th60;
    hence thesis;
  end;
  then A c= the carrier of Centralizer H;
  hence A is Subset of Centralizer H;
end;
