
theorem Th33:
  for k being Element of NAT, a being non empty Element of
SubstPoset (NAT, {k}) st a <> {{}} ex f being finite Function st f in a & f <>
  {}
proof
  let k be Element of NAT;
  let a be non empty Element of SubstPoset (NAT, {k});
  assume
A1: a <> {{}};
  consider f being object such that
A2: f in a by XBOOLE_0:def 1;
  SubstitutionSet (NAT, {k}) = the carrier of SubstPoset (NAT, {k}) by
SUBSTLAT:def 4;
  then reconsider f as finite Function by A2,HEYTING2:1;
  take f;
  thus f in a by A2;
  assume f = {};
  hence thesis by A1,A2,Th32;
end;
