
theorem Th25:
  for L being distributive LATTICE, I being Ideal of L, F being
  Filter of L st I misses F ex P being Filter of L st P is prime & F c= P & I
  misses P
proof
  let L be distributive LATTICE, I be Ideal of L, F be Filter of L such that
A1: I misses F;
  reconsider I9 = F as Ideal of L opp by YELLOW_7:27,29;
  reconsider F9 = I as Filter of L opp by YELLOW_7:26,28;
  consider P9 being Ideal of L opp such that
A2: P9 is prime & I9 c= P9 & P9 misses F9 by A1,Th23;
  reconsider P = P9 as Filter of L by YELLOW_7:27,29;
  take P;
  thus thesis by A2,Th17;
end;
