
theorem PrimFil:
  for L being distributive Lattice holds
    F_primeSet L c< PFilters L
  proof
    let L be distributive Lattice;
A1: F_primeSet L c= PFilters L
    proof
      let x be object;
      assume x in F_primeSet L; then
      consider F being Filter of L such that
A2:   x = F & F <> the carrier of L & F is prime;
      thus thesis by A2;
    end;
    not <.L.) in F_primeSet L
    proof
      assume <.L.) in F_primeSet L; then
      consider F being Filter of L such that
B1:   F = <.L.) & F <> the carrier of L & F is prime;
      thus thesis by B1;
    end;
    hence thesis by A1;
  end;
