
theorem Th40:
  for m being Element of NAT, a being Element of SubstPoset (NAT,
  {m}) st PFDrt m is_>=_than a holds a <> {{}}
proof
  let m be Element of NAT;
  reconsider P1 = PFBrt (1,m) as Element of SubstPoset (NAT, {m}) by Th25;
  let a be Element of SubstPoset (NAT, {m});
  assume
A1: PFDrt m is_>=_than a;
  PFBrt (1,m) in PFDrt m by Def5;
  then
A2: P1 >= a by A1;
  assume
A3: a = {{}};
  Top SubstPoset (NAT, {m}) = {{}} by Th36;
  hence thesis by A3,A2,Th27,WAYBEL15:3;
end;
