
theorem Th67:
  for T being adj-structured antisymmetric non void reflexive
transitive with_suprema Noetherian TAS-structure for t1,t2 being type of T st
  T@--> reduces t1,t2 holds t1 <= t2
proof
  let T be adj-structured with_suprema antisymmetric non empty non void
  reflexive transitive Noetherian TAS-structure;
  let t1,t2 be type of T;
  set R = T@-->;
  defpred P[Element of T, Element of T] means $1 <= $2;
A1: for x,y,z be Element of T holds P[x, y] & P[y, z] implies P[x, z] by
YELLOW_0:def 2;
A2: now
    let x,y be Element of T;
    R c= the InternalRel of T by Th66;
    hence [x,y] in R implies P[x, y] by ORDERS_2:def 5;
  end;
A3: for x being Element of T holds P[x, x];
  for x,y being Element of T st R reduces x,y holds P[x,y] from RedInd(A2,
  A3,A1);
  hence thesis;
end;
