
theorem
  for L being with_infima Poset, I being Ideal of L holds I is prime iff
  I` is Filter of L or I` = {}
proof
  let L be with_infima Poset, I be Ideal of L;
  set F = I`;
  thus I is prime implies I` is Filter of L or I` = {}
  proof
    assume
A1: for x,y being Element of L st x"/\"y in I holds x in I or y in I;
A2: F is filtered
    proof
      let x,y be Element of L;
      assume x in F & y in F;
      then ( not x in I)& not y in I by XBOOLE_0:def 5;
      then
A3:   x"/\"y in F by A1,SUBSET_1:29;
      x"/\"y <= x & x"/\" y <= y by YELLOW_0:23;
      hence thesis by A3;
    end;
    F is upper
    proof
      let x,y be Element of L;
      assume that
A4:   x in F and
A5:   y >= x;
      y in I implies x in I by A5,WAYBEL_0:def 19;
      hence thesis by A4,XBOOLE_0:def 5;
    end;
    hence thesis by A2;
  end;
  assume
A6: I` is Filter of L or I` = {};
  let x,y be Element of L;
  assume x"/\"y in I;
  then not x"/\"y in F by XBOOLE_0:def 5;
  then not x in F or not y in F by A6,WAYBEL_0:41;
  hence thesis by SUBSET_1:29;
end;
