
theorem Th22:
  for L being Boolean LATTICE, F being Filter of L holds F is
  proper prime iff F is ultra
proof
  let L be Boolean LATTICE;
  let F be Filter of L;
  thus F is proper prime implies F is ultra
  proof
    assume
A1: F is proper;
    assume
A2: for x,y being Element of L st x"\/"y in F holds x in F or y in F;
    thus F is proper by A1;
    let G be Filter of L;
    assume that
A3: F c= G and
A4: F <> G;
    consider x being object such that
A5: not (x in F iff x in G) by A4,TARSKI:2;
    reconsider x as Element of L by A5;
    set y = 'not' x;
    x"\/"y = Top L by YELLOW_5:34;
    then y in F by A2,A3,A5,WAYBEL_4:22;
    then x"/\"y in G by A3,A5,WAYBEL_0:41;
    then
A6: Bottom L in G by YELLOW_5:34;
    thus G c= the carrier of L;
    let x be object;
    assume x in the carrier of L;
    then reconsider x as Element of L;
    x >= Bottom L by YELLOW_0:44;
    hence thesis by A6,WAYBEL_0:def 20;
  end;
  assume that
A7: F is proper and
A8: for G being Filter of L st F c= G holds F = G or G = the carrier of L;
  thus F is proper by A7;
  now
    let a be Element of L;
    assume that
A9: not a in F and
A10: not 'not' a in F;
    set b = 'not' a;
A11: F \/ {a} c= uparrow fininfs (F \/ { a}) by WAYBEL_0:62;
    {a} c= F \/ {a} & a in {a} by TARSKI:def 1,XBOOLE_1:7;
    then a in F \/ {a};
    then F c= F \/ {a} & a in uparrow fininfs (F \/ {a}) by A11,XBOOLE_1:7;
    then uparrow fininfs (F \/ {a}) = the carrier of L by A8,A9,A11,XBOOLE_1:1;
    then consider c being Element of L such that
A12: c in F and
A13: b >= c "/\" inf {a} by Lm1;
    c <= Top L by YELLOW_0:45;
    then
A14: c = c"/\"Top L by YELLOW_0:25
      .= c"/\"(a"\/"b) by YELLOW_5:34
      .= (c"/\"a) "\/" (c"/\"b) by WAYBEL_1:def 3;
    inf {a} = a & c"/\"b <= b by YELLOW_0:23,39;
    then c <= b by A13,A14,YELLOW_0:22;
    hence contradiction by A10,A12,WAYBEL_0:def 20;
  end;
  hence thesis by Th20;
end;
