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
  holds Centralizer (Omega).G = center G
proof
  let G be Group;
  for g being Element of G holds g in Centralizer (Omega).G iff g in center G
  proof
    let g be Element of G;
    thus g in Centralizer (Omega).G implies g in center G
    proof
      assume A1: g in Centralizer (Omega).G;
      for a being Element of G holds g*a = a*g
      proof
        let a be Element of G;
        a in (Omega).G;
        hence g*a = a*g by A1,Th60;
      end;
      hence g in center G by GROUP_5:77;
    end;
    thus g in center G implies g in Centralizer (Omega).G
    proof
      assume g in center G;
      then for b being Element of G st b in (Omega).G holds g*b = b*g
      by GROUP_5:77;
      then g is Element of Centralizer (Omega).G by Th60;
      hence thesis;
    end;
  end;
  hence Centralizer (Omega).G = center G;
end;
