
theorem Th26:
  for T being adj-structured reflexive transitive antisymmetric
  with_suprema TA-structure for A being Subset of the adjectives of T for t
  being type of T st A is_applicable_to t holds types A /\ downarrow t is Ideal
  of T
proof
  let T be adj-structured reflexive transitive antisymmetric with_suprema
  TA-structure;
  let A be Subset of the adjectives of T;
  let t be type of T;
  given t9 being type of T such that
A1: A c= adjs t9 and
A2: t9 <= t;
A3: t9 in downarrow t by A2,WAYBEL_0:17;
  t9 in types A by A1,Th14;
  hence thesis by A3,WAYBEL_0:27,44,XBOOLE_0:def 4;
end;
