
theorem Th30:
  for m being Element of NAT, a being Element of SubstPoset (NAT,
  {m}) st PFDrt m is_>=_than a holds for X being non empty set st X in a holds
  not ( for n being Element of NAT st [n,m] in X holds n is odd )
proof
  let m be Element of NAT;
  let a be Element of SubstPoset (NAT, {m});
  assume
A1: PFDrt m is_>=_than a;
  let X be non empty set;
  assume
A2: X in a;
  then reconsider X9= X as finite non empty Subset of [:NAT,{m}:] by Th29;
  assume
A3: for n being Element of NAT st [n,m] in X holds n is odd;
A4: now
    let k be non zero Element of NAT;
    reconsider Pk = PFBrt (k,m) as Element of SubstPoset (NAT, {m}) by Th25;
A5: [2*k+1,m] in PFCrt (k,m) by Def3;
    Pk in PFDrt m by Def5;
    then a <= Pk by A1;
    then consider y being set such that
A6: y in Pk and
A7: y c= X by A2,Th12;
A8: not ex m9 being Element of NAT st m9 <= k & y = PFArt (m9,m)
    proof
      given m9 being Element of NAT such that
      m9 <= k and
A9:   y = PFArt (m9,m);
      [2*m9,m] in PFArt (m9,m) by Def2;
      hence thesis by A3,A7,A9;
    end;
    ( ex m9 being non zero Element of NAT st m9 <= k & y = PFArt (m9,m) )
    or y = PFCrt (k,m) by A6,Def4;
    hence [2*k+1,m] in X by A7,A8,A5;
  end;
  ex l being non zero Element of NAT st not [2*l+1,m] in X9 by Th4;
  hence thesis by A4;
end;
