
theorem Th20:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema TA-structure for t being type of T for a being
  adjective of T st a is_applicable_to t holds a ast t <= t
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema TA-structure;
  let t be type of T;
  let a be adjective of T;
  assume a is_applicable_to t;
  then types a /\ downarrow t is Ideal of T by Th19;
  then sup (types a /\ downarrow t) in types a /\ downarrow t by Th1;
  then a ast t in downarrow t by XBOOLE_0:def 4;
  hence thesis by WAYBEL_0:17;
end;
