
theorem Th34:
  for k being Element of NAT, a being non empty Element of
SubstPoset (NAT, {k}), a9 being Element of Fin PFuncs (NAT, {k}) st a <> {{}} &
  a = a9 holds Involved a9 is finite non empty Subset of NAT
proof
  let k be Element of NAT;
  let a be non empty Element of SubstPoset (NAT, {k});
  let a9 be Element of Fin PFuncs (NAT, {k});
  assume that
A1: a <> {{}} and
A2: a = a9;
  consider f being finite Function such that
A3: f in a and
A4: f <> {} by A1,Th33;
  ex k1 being object st k1 in dom f by A4,XBOOLE_0:def 1;
  hence thesis by A2,A3,HEYTING2:6,def 1;
end;
