
theorem Th19:
  for T being adj-structured reflexive transitive antisymmetric
with_suprema TA-structure for a being adjective 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 adjective of T;
  let t be type of T;
  given t9 being type of T such that
A1: t9 in types a and
A2: t9 <= t;
  t9 in downarrow t by A2,WAYBEL_0:17;
  hence thesis by A1,WAYBEL_0:27,44,XBOOLE_0:def 4;
end;
