
theorem Th35:
  for L being continuous LATTICE for p being Element of L st p is
  pseudoprime for A being finite non empty Subset of L st inf A << p ex a being
  Element of L st a in A & a <= p
proof
  let L be continuous LATTICE;
  let p be Element of L;
  given P being prime Ideal of L such that
A1: p = sup P;
  let A be finite non empty Subset of L;
  assume inf A << p;
  then
A2: inf A in waybelow p;
  waybelow p c= P by A1,WAYBEL_5:1;
  then consider a being Element of L such that
A3: a in A & a in P by A2,Th12;
  take a;
  ex_sup_of P,L by WAYBEL_0:75;
  then P is_<=_than p by A1,YELLOW_0:30;
  hence thesis by A3,LATTICE3:def 9;
end;
