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

theorem Th23:
  for p be Element of OrderedNAT holds
  {x where x is Element of NAT:ex p0 be Element of NAT st
  p=p0 & p0 <= x}=uparrow p
  proof
    let p be Element of OrderedNAT;
    reconsider p0=p as Element of NAT;
A1: for p be Element of the carrier of OrderedNAT holds
    {x where x is Element of the carrier of OrderedNAT:p <= x}=uparrow p
    proof
      let p be Element of OrderedNAT;
      hereby
        let t be object;
        assume t in {x where x is Element of OrderedNAT:p <=x};
        then consider x0 be Element of OrderedNAT such that
A2:     t=x0 and
A3:     p <= x0;
        thus t in uparrow p by A2,A3,WAYBEL_0:18;
      end;
      let t be object;
      assume
A4:   t in uparrow p;
      then reconsider t0=t as Element of OrderedNAT;
      p <= t0 by A4,WAYBEL_0:18;
      hence t in {x where x is Element of OrderedNAT:p <=x};
    end;
    {x where x is Element of NAT:ex p0 be Element of NAT st
    p=p0 & p0 <= x} = {x where x is Element of OrderedNAT:p <= x} by Lm4;
    hence thesis by A1;
  end;
