reserve X for set,
  F for Filter of BoolePoset X,
  x for Element of BoolePoset X ,
  z for Element of X;

theorem Th10:
  card FixedUltraFilters X = card X
proof
  set FUF = { uparrow x : ex z st x = {z} };
A1: BoolePoset X = InclPoset bool X by YELLOW_1:4;
A2: InclPoset bool X = RelStr(#bool X, RelIncl bool X#) by YELLOW_1:def 1;
  then
A3: the carrier of BoolePoset X = bool X by YELLOW_1:4;
  X,FUF are_equipotent
  proof
    defpred P[object, object] means
ex y being Element of X, x being Element of
    BoolePoset X st x = {y} & $1 = y & $2 = uparrow x;
A4: for x being object st x in X ex y being object st P[x,y]
    proof
      let x be object;
      assume
A5:   x in X;
      then reconsider x9 = x as Element of X;
      reconsider bx = {x} as Element of BoolePoset X by A1,A2,A5,ZFMISC_1:31;
      take uparrow bx;
      take x9;
      take bx;
      thus thesis;
    end;
    consider f being Function such that
A6: dom f = X & for x being object st x in X holds P[x, f.x] from
    CLASSES1:sch 1(A4);
    take f;
    thus f is one-to-one
    proof
      let x1, x2 be object such that
A7:   x1 in dom f and
A8:   x2 in dom f and
A9:   f.x1 = f.x2;
      consider x29 being Element of X, bx2 being Element of BoolePoset X such
      that
A10:  bx2 = {x29} & x2 = x29 and
A11:  f.x2 = uparrow bx2 by A6,A8;
      consider x19 being Element of X, bx1 being Element of BoolePoset X such
      that
A12:  bx1 = {x19} & x1 = x19 and
A13:  f.x1 = uparrow bx1 by A6,A7;
      bx1 = bx2 by A9,A13,A11,WAYBEL_0:20;
      hence thesis by A12,A10,ZFMISC_1:3;
    end;
    thus dom f = X by A6;
    now
      let z be object;
      hereby
        assume z in rng f;
        then consider x1 being object such that
A14:    x1 in dom f and
A15:    z = f.x1 by FUNCT_1:def 3;
        ex x19 being Element of X, bx1 being Element of BoolePoset X st
        bx1 = {x19} & x1 = x19 & f.x1 = uparrow bx1 by A6,A14;
        hence z in FUF by A15;
      end;
      assume z in FUF;
      then consider bx being Element of BoolePoset X such that
A16:  z = uparrow bx and
A17:  ex y being Element of X st bx = {y};
      consider y being Element of X such that
A18:  bx = {y} by A17;
A19:  y in X by A3,A18,ZFMISC_1:31;
      then ex x19 being Element of X, bx1 being Element of BoolePoset X st bx1
      = {x19} & y = x19 & f.y = uparrow bx1 by A6;
      hence z in rng f by A6,A16,A18,A19,FUNCT_1:def 3;
    end;
    hence thesis by TARSKI:2;
  end;
  hence thesis by CARD_1:5;
end;
