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
  center G is characteristic Subgroup of G
proof
  set Z = center G;
  A1: for y,z being Element of G st z in Z holds (phi.z)*y = y*phi.z
  proof
    let y,z be Element of G;
    assume B1: z in Z;
    set x = (phi").y;
    (phi.z)*y = (phi.z)*(phi.x) by Th4
             .= phi.(z*x) by GROUP_6:def 6
             .= phi.(x*z) by B1,GROUP_5:77
             .= (phi.x)*(phi.z) by GROUP_6:def 6
             .= y*(phi.z) by Th4;
    hence thesis;
  end;
  A2: for z being Element of G st z in Z holds (phi|Z).z in Z
  proof
    let z be Element of G;
    assume B1: z in Z;
    then for y being Element of G holds (phi.z)*y=y*(phi.z) by A1;
    then (phi.z) in Z by GROUP_5:77;
    hence ((phi|Z).z) in Z by B1,Th1;
  end;
  Image(phi|Z) is Subgroup of Z
  proof
    for w being Element of G st w in rng(phi|Z) holds w in Z
    proof
      let w be Element of G;
      assume w in rng(phi|Z);
      then consider z being object such that
      C1: z in dom(phi|Z) and
      C2: (phi|Z).z = w by FUNCT_1:def 3;
      reconsider z as Element of Z by C1;
      z is Element of G by GROUP_2:42;
      hence w in Z by C2,A2,STRUCT_0:def 5;
    end;
    then rng(phi|Z) c= the carrier of Z by STRUCT_0:def 5;
    then the carrier of Image(phi|Z) c= the carrier of Z by GROUP_6:44;
    hence Image(phi|Z) is Subgroup of Z by GROUP_2:57;
  end;
  hence Z is characteristic Subgroup of G by Th40;
end;
