
theorem Th68:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric non void with_suprema TAS-structure holds T@--> is irreflexive
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric non
  void with_suprema TAS-structure;
  set R = T@-->;
  let x be object;
  assume x in field R;
  assume
A1: [x,x] in R;
  then reconsider x as type of T by ZFMISC_1:87;
  consider a being adjective of T such that
A2: not a in adjs x and
A3: a is_properly_applicable_to x and
A4: a ast x = x by A1,Def31;
  a is_applicable_to x by A3;
  hence thesis by A2,A4,Th21;
end;
