
theorem Th32:
  for n being Element of NAT, a being Element of SubstPoset (NAT,
  {n}) st {} in a holds a = {{}}
proof
  let n be Element of NAT;
  let a be Element of SubstPoset (NAT, {n});
  assume
A1: {} in a;
  SubstitutionSet (NAT, {n}) = the carrier of SubstPoset (NAT, {n}) by
SUBSTLAT:def 4;
  then
A2: a in SubstitutionSet (NAT, {n});
  assume a <> {{}};
  then ex k being set st k in a & k <> {} by A1,Lm9;
  hence thesis by A2,A1,SUBSTLAT:5,XBOOLE_1:2;
end;
