
theorem Th81:
  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, a being adjective of T st a
  is_properly_applicable_to t1 & a ast t1 <= radix t2 holds t1 <= radix t2
proof
  let T be adj-structured with_suprema antisymmetric commutative non empty
  non void reflexive transitive Noetherian TAS-structure;
  let t1,t2 be type of T, a be adjective of T;
  set R = T@-->;
  set AA = {t9 where t9 is type of T: ex A being finite Subset of the
  adjectives of T st A is_properly_applicable_to t9 & A ast t9 = t2};
  assume that
A1: a is_properly_applicable_to t1 and
A2: a ast t1 <= radix t2;
  consider A being finite Subset of the adjectives of T such that
A3: A is_properly_applicable_to t1 "\/" radix t2 and
A4: A ast (t1 "\/" radix t2) = radix t2 by A1,A2,Def30;
  consider v1 being FinSequence of the adjectives of T such that
A5: rng v1 = A and
A6: v1 is_properly_applicable_to t1 "\/" radix t2 by A3;
  R is with_Church-Rosser_property with_UN_property strongly-normalizing
  Relation by Th69,Th77;
  then nf(t2, R) is_a_normal_form_of t2, R by REWRITE1:54;
  then R reduces t2, nf(t2,R);
  then consider B being finite Subset of the adjectives of T such that
A7: B is_properly_applicable_to radix t2 and
A8: t2 = B ast radix t2 by Th76;
  consider v2 being FinSequence of the adjectives of T such that
A9: rng v2 = B and
A10: v2 is_properly_applicable_to radix t2 by A7;
A11: rng (v1^v2) = A \/ B by A5,A9,FINSEQ_1:31;
A12: radix t2 = v1 ast (t1 "\/" radix t2) by A4,A5,A6,Th56,Th57;
  then
A13: v1^v2 is_properly_applicable_to t1 "\/" radix t2 by A6,A10,Th61;
  then
A14: A \/ B is_properly_applicable_to t1 "\/" radix t2 by A11;
  (A \/ B) ast (t1 "\/" radix t2) = v1^v2 ast (t1 "\/" radix t2) by A11,A13
,Th56,Th57
    .= v2 ast radix t2 by A12,Th37
    .= t2 by A8,A9,A10,Th56,Th57;
  then t1 "\/" radix t2 in AA by A14;
  then t1 "\/" radix t2 <= "\/"(AA, T) by Th80,YELLOW_4:1;
  then
A15: t1 "\/" radix t2 <= radix t2 by Th80;
  t1 "\/" radix t2 >= t1 by YELLOW_0:22;
  hence thesis by A15,YELLOW_0:def 2;
end;
