
theorem Th52:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema TA-structure for t being type of T for A being
Subset of the adjectives of T for t9 being type of T st t9 <= t & A c= 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 Subset of the adjectives of T;
  let t9 be type of T;
  assume that
A1: t9 <= t and
A2: a c= adjs t9;
A3: t9 in downarrow t by A1,WAYBEL_0:17;
  thus a is_applicable_to t
  by A1,A2;
  then types a /\ downarrow t is Ideal of T by Th26;
  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,Th14;
  then t9 in types a /\ downarrow t by A3,XBOOLE_0:def 4;
  hence thesis by A4;
end;
