reserve x,y for set,
  k,n for Nat,
  i for Integer,
  G for Group,
  a,b,c ,d,e for Element of G,
  A,B,C,D for Subset of G,
  H,H1,H2,H3,H4 for Subgroup of G ,
  N1,N2 for normal Subgroup of G,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem
  center G is normal Subgroup of G
proof
  now
    let a;
    thus a * center G c= center G * a
    proof
      let x be object;
      assume x in a * center G;
      then consider b such that
A1:   x = a * b and
A2:   b in center G by GROUP_2:103;
      x = b * a by A1,A2,Th77;
      hence thesis by A2,GROUP_2:104;
    end;
  end;
  hence thesis by GROUP_3:118;
end;
