reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th21:
  ex F be sequence of bool NAT st
  for x be Element of NAT holds F.x = {y where y is Element of NAT:x <= y}
  proof
    deffunc FF(object)={y where y is Element of NAT:ex x0 be Element of NAT
    st x0=$1 & x0 <= y};
A1: now
      let x be object;
      assume x in NAT;
      now
        let t be object;
        assume t in {y where y is Element of NAT:ex x0 be Element of NAT st
        x0=x & x0 <= y};
        then consider y0 be Element of NAT such that
A2:     t=y0 and
        ex x0 be Element of NAT st x0=x & x0 <=y0;
        thus t in NAT by A2;
      end;
      then {y where y is Element of NAT:ex x0 be Element of NAT st
      x0=x & x0 <= y} c= NAT;
      hence FF(x) in bool NAT;
    end;
    ex f being Function of NAT,bool NAT st
    for x being object st x in NAT holds f.x=FF(x) from FUNCT_2:sch 2(A1);
    then consider F being Function of NAT,bool NAT such that
A3: for x being object st x in NAT holds F.x=FF(x);
    for x be Element of NAT holds F.x = {y where y is Element of NAT:x <=y}
    proof
      let x be Element of NAT;
      {y where y is Element of NAT:ex x0 be Element of NAT st x0=x & x0 <= y}=
      {y where y is Element of NAT:x <= y}
      proof
        hereby let t be object;assume t in
          {y where y is Element of NAT:ex x0 be Element of NAT st x0=x &
          x0 <= y};
          then consider y0 be Element of NAT such that
A4:       t=y0 and
A5:       ex x0 be Element of NAT st x0=x & x0 <= y0;
          consider x1 be Element of NAT such that
A6:       x1=x and
A7:       x1 <= y0 by A5;
          thus t in {y where y is Element of NAT:x <= y} by A4,A6,A7;
        end;
        let t be object;assume t in {y where y is Element of NAT:x <= y};
        then consider y0 be Element of NAT such that
A8:     t=y0 and
A9:     x <= y0;
        thus t in {y where y is Element of NAT:ex x0 be Element of NAT st
        x0=x & x0 <= y} by A8,A9;
      end;
      hence thesis by A3;
    end;
    hence thesis;
  end;
