
theorem Th66:
  for T being adj-structured antisymmetric non void reflexive
  transitive with_suprema Noetherian TAS-structure holds T@--> c= the
  InternalRel of T
proof
  let T be adj-structured with_suprema antisymmetric non empty non void
  reflexive transitive Noetherian TAS-structure;
  let t1,t2 be Element of T;
  reconsider q1 = t1, q2 = t2 as type of T;
  assume [t1,t2] in T@-->;
  then consider a being adjective of T such that
  not a in adjs q2 and
A1: a is_properly_applicable_to q2 and
A2: a ast q2 = q1 by Def31;
  a is_applicable_to q2 by A1;
  then q1 <= q2 by A2,Th20;
  hence thesis by ORDERS_2:def 5;
end;
