
theorem Th37:
  for k being Element of NAT holds Bottom 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:1;
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 {};
    for x be set st x in a ex y being set st y in b & y c= x;
    hence thesis by Th12;
  end;
  a is_>=_than {};
  then a = "\/"({},SubstPoset (NAT, {k})) by A1,YELLOW_0:30;
  hence thesis by YELLOW_0:def 11;
end;
