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
  for H being Subgroup of G holds H is Subgroup of center G implies H is
  normal Subgroup of G
proof
  let H be Subgroup of G;
  assume
A1: H is Subgroup of center G;
  now
    let a;
    thus H * a c= a * H
    proof
      let x be object;
      assume x in H * a;
      then consider b such that
A2:   x = b * a and
A3:   b in H by GROUP_2:104;
      b in center G by A1,A3,GROUP_2:40;
      then x = a * b by A2,Th77;
      hence thesis by A3,GROUP_2:103;
    end;
  end;
  hence thesis by GROUP_3:119;
end;
