
theorem Th20:
  for L being Boolean LATTICE, F being Filter of L holds F is
  prime iff for a being Element of L holds a in F or 'not' a in F
proof
  let L be Boolean LATTICE;
  let F be Filter of L;
  hereby
    assume
A1: F is prime;
    let a be Element of L;
    set b = 'not' a;
    a"\/"b = Top L by YELLOW_5:34;
    then a"\/"b in F by WAYBEL_4:22;
    hence a in F or b in F by A1;
  end;
  assume
A2: for a being Element of L holds a in F or 'not' a in F;
  let a,b be Element of L;
  assume that
A3: a"\/"b in F and
A4: not a in F and
A5: not b in F;
  'not' a in F & 'not' b in F by A2,A4,A5;
  then ('not' a)"/\"'not' b in F by WAYBEL_0:41;
  then 'not' (a"\/"b) in F by YELLOW_5:36;
  then 'not' (a"\/"b)"/\"(a"\/"b) in F by A3,WAYBEL_0:41;
  then
A6: Bottom L in F by YELLOW_5:34;
  a >= Bottom L by YELLOW_0:44;
  hence contradiction by A4,A6,WAYBEL_0:def 20;
end;
