
theorem Th69:
  for T being adj-structured antisymmetric non void reflexive
  transitive with_suprema Noetherian TAS-structure holds T@--> is
  strongly-normalizing
proof
  let T be adj-structured with_suprema antisymmetric non empty non void
  reflexive transitive Noetherian TAS-structure;
  set R = T@-->, Q = the InternalRel of T;
A1: field R c= field Q by Th66,RELAT_1:16;
A2: R c= Q by Th66;
  R is co-well_founded
  proof
    let Y be set;
    assume that
A3: Y c= field R and
A4: Y <> {};
    Y c= field Q by A1,A3;
    then consider a being object such that
A5: a in Y and
A6: for b being object st b in Y & a <> b holds not [a,b] in Q by A4,
REWRITE1:def 16;
    take a;
    thus thesis by A2,A5,A6;
  end;
  then R is irreflexive co-well_founded Relation by Th68;
  hence thesis;
end;
