
theorem Th36:
  for k being Element of NAT holds Top SubstPoset (NAT, {k}) = {{} }
proof
  let k be Element of NAT;
  SubstitutionSet (NAT, {k}) = the carrier of SubstPoset (NAT, {k}) by
SUBSTLAT:def 4;
  then reconsider a = {{}} as Element of SubstPoset (NAT, {k}) by SUBSTLAT:2;
A1: for b being Element of SubstPoset (NAT, {k}) st b is_<=_than {} holds a
  >= b
  proof
    let b be Element of SubstPoset (NAT, {k});
    assume b is_<=_than {};
    now
      let x be set;
      assume x in b;
      take y = {};
      thus y in a & y c= x by TARSKI:def 1;
    end;
    hence thesis by Th12;
  end;
  a is_<=_than {};
  then a = "/\"({},SubstPoset (NAT, {k})) by A1,YELLOW_0:31;
  hence thesis by YELLOW_0:def 12;
end;
