
theorem
  for T being adj-structured with_suprema antisymmetric commutative non
  empty non void reflexive transitive Noetherian TAS-structure for t1,t2 being
  type of T st t1 <= t2 holds radix t1 <= radix t2
proof
  let T be adj-structured with_suprema antisymmetric commutative non empty
  non void reflexive transitive Noetherian TAS-structure;
  set R = T@-->;
  let t1, t2 be type of T such that
A1: t1 <= t2;
  t2 <= radix t2 by Th67,Th78;
  then
A2: t1 <= radix t2 by A1,YELLOW_0:def 2;
  set X = the carrier of T;
  defpred P[Element of X, Element of X] means $1 <= radix t2 implies $2 <=
  radix t2;
A3: for x,y,z being Element of X st P[x,y] & P[y,z] holds P[x,z];
A4: now
    let x,y be Element of X;
    reconsider t1 = x, t2 = y as type of T;
    assume [x,y] in R;
    then ex a being adjective of T st not a in adjs t2 & a
    is_properly_applicable_to t2 & a ast t2 = t1 by Def31;
    hence P[x,y] by Th81;
  end;
A5: for x being Element of X holds P[x,x];
  for x,y being Element of T st R reduces x,y holds P[x,y] from RedInd(A4,
  A5,A3);
  hence thesis by A2,Th78;
end;
