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

theorem OperLat:
  for n being non zero Nat,
      a, b being Element of Divisors_Lattice n holds
    a "\/" b = a lcm b &
    a "/\" b = a gcd b
  proof
    let n be non zero Nat,
        a, b be Element of Divisors_Lattice n;
    set L = Divisors_Lattice n;
    set K = NatPlus_Lattice;
A0: the carrier of L c= the carrier of K by NAT_LAT:def 12;
    reconsider aa = a, bb = b as Element of K by A0;
    the L_join of L = (the L_join of K) ||
      the carrier of L by NAT_LAT:def 12;
    hence a "\/" b = aa "\/" bb by FUNCT_1:49,ZFMISC_1:87
            .= a lcm b;
    the L_meet of L = (the L_meet of K) ||
      the carrier of L by NAT_LAT:def 12;
    hence a "/\" b = aa "/\" bb by FUNCT_1:49,ZFMISC_1:87
            .= a gcd b;
  end;
