
theorem Th23:
  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 for t9 being type of T st t9 <= t & a in adjs t9 holds a
  is_applicable_to t & t9 <= a ast 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;
  let t9 be type of T;
  assume that
A1: t9 <= t and
A2: a in adjs t9;
A3: t9 in downarrow t by A1,WAYBEL_0:17;
  thus a is_applicable_to t
  by A1,A2,Th13;
  then types a /\ downarrow t is Ideal of T by Th19;
  then ex_sup_of types a /\ downarrow t, T by Th1;
  then
A4: a ast t is_>=_than types a /\ downarrow t by YELLOW_0:30;
  t9 in types a by A2,Th13;
  then t9 in types a /\ downarrow t by A3,XBOOLE_0:def 4;
  hence thesis by A4;
end;
