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 Th59:
  for G being Group
  for H being Subgroup of G
  holds the carrier of Centralizer H = {b where b is Element of G : for a
  being Element of G st a in H holds b*a=a*b}
proof
  let G be Group;
  let H be Subgroup of G;
  set A = carr H;
  set Car = {b where b is Element of G : for a being Element of G st a in H
                                         holds b*a=a*b};
  A1: the carrier of Centralizer A = {b where b is Element of G : for a
  being Element of G st a in A holds b*a=a*b} by Def4;
  for x being object st x in Car holds x in the carrier of Centralizer H
  proof
    let x be object;
    assume B1: x in Car;
    ex b being Element of G
    st x = b & (for a being Element of G st a in carr(H) holds b*a=a*b)
    proof
      consider b being Element of G such that
      B2: x = b and
      B3: for a being Element of G st a in H holds b*a=a*b
      by B1;
      for a being Element of G st a in carr H holds b*a=a*b
      proof
        let a be Element of G;
        assume a in carr H;
        then a in H;
        hence b*a=a*b by B3;
      end;
      hence thesis by B2;
    end;
    then x in the carrier of Centralizer carr H by A1;
    hence thesis by Def5;
  end;
  then A3: Car c= the carrier of Centralizer H;

  for x being object st x in the carrier of Centralizer H holds x in Car
  proof
    let x be object;
    assume x in the carrier of Centralizer H;
    then B1: x in the carrier of Centralizer carr H by Def5;
    ex b being Element of G
    st (x=b) & (for a being Element of G st a in H holds b*a=a*b)
    proof
      consider b being Element of G such that
      Z1: x = b  & (for a being Element of G st a in carr H holds b*a=a*b)
      by A1,B1;
      for a being Element of G st a in H holds b*a=a*b by Z1;
      hence thesis by Z1;
    end;
    hence x in Car;
  end;
  then the carrier of Centralizer H c= Car;
  hence thesis by A3,XBOOLE_0:def 10;
end;
