 reserve n,i for Nat;
 reserve p for Prime;

theorem TopBot:
  for n being non zero Nat holds
    Top Divisors_Lattice n = n &
    Bottom Divisors_Lattice n = 1
  proof
    let n be non zero Nat;
    set L = Divisors_Lattice n;
    n in NatDivisors n by MOEBIUS1:39; then
    reconsider TT = n as Element of L by DivLat;
a1: for a being Element of L holds TT "\/" a = TT & a "\/" TT = TT
    proof
      let a be Element of L;
      a in the carrier of L; then
      a in NatDivisors n by DivLat;
      hence thesis by NEWTON:44,MOEBIUS1:39;
    end;
    1 in NatDivisors n by MOEBIUS1:39,NAT_D:6; then
    reconsider TT = 1 as Element of L by DivLat;
    for a being Element of L holds TT "/\" a = TT & a "/\" TT = TT
      by NEWTON:51;
    hence thesis by a1,LATTICES:def 16,def 17;
  end;
