
theorem Th24:
  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 in adjs t holds a is_applicable_to t & 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
A1: a in adjs t;
  hence a is_applicable_to t by Th23;
  then
A2: a ast t <= t by Th20;
  t <= a ast t by A1,Th23;
  hence thesis by A2,YELLOW_0:def 3;
end;
