
theorem
  for L being distributive continuous LATTICE for p being Element of L
  st uparrow fininfs (downarrow p)` misses waybelow p holds p is pseudoprime
proof
  let L be distributive continuous LATTICE;
  let p be Element of L;
  set I = waybelow p;
  set F = uparrow fininfs (downarrow p)`;
A1: ex_sup_of downarrow p, L & sup downarrow p = p by WAYBEL_0:34;
  (downarrow p)` c= F by WAYBEL_0:62;
  then
A2: F` c= (downarrow p)`` by SUBSET_1:12;
  assume F misses I;
  then consider P being Ideal of L such that
A3: P is prime and
A4: I c= P and
A5: P misses F by Th23;
  reconsider P as prime Ideal of L by A3;
A6: ex_sup_of P, L by WAYBEL_0:75;
  ex_sup_of I, L & p = sup I by WAYBEL_0:75,WAYBEL_3:def 5;
  then
A7: sup P >= p by A4,A6,YELLOW_0:34;
  take P;
  P c= F` by A5,SUBSET_1:23;
  then sup P <= p by A6,A2,A1,XBOOLE_1:1,YELLOW_0:34;
  hence thesis by A7,ORDERS_2:2;
end;
