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,B being Subset of G
  st A c= B
  holds Centralizer B is Subgroup of Centralizer A
proof
  let G be Group;
  let A,B be Subset of G;
  assume A1: A c= B;
  for g being Element of G st g in Centralizer B
  holds g in Centralizer A
  proof
    let g be Element of G;
    assume g in Centralizer B;
    then for a being Element of G st a in A
    holds g*a = a*g by A1,Th57;
    then g is Element of Centralizer A by Th57;
    hence thesis;
  end;
  hence Centralizer B is Subgroup of Centralizer A by GROUP_2:58;
end;
