
theorem Th21:
  for X being set, F being Filter of BoolePoset X holds F is prime
  iff for A being Subset of X holds A in F or X\A in F
proof
  let X be set;
  set L = BoolePoset X;
  let F be Filter of L;
  L = InclPoset bool X by YELLOW_1:4;
  then
A1: L = RelStr(#bool X, RelIncl bool X#) by YELLOW_1:def 1;
  hereby
    assume
A2: F is prime;
    let A be Subset of X;
    reconsider a = A as Element of L by A1;
    a in F or 'not' a in F by A2,Th20;
    hence A in F or X\A in F by Th5;
  end;
  assume
A3: for A being Subset of X holds A in F or X\A in F;
  now
    let a be Element of L;
    'not' a = X\a by Th5;
    hence a in F or 'not' a in F by A1,A3;
  end;
  hence thesis by Th20;
end;
