
theorem Th80:
  for T being adj-structured with_suprema antisymmetric
commutative non empty non void reflexive transitive Noetherian TAS-structure
  for t being type of T for X being set st X = {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 = t} holds ex_sup_of X, T & radix t = "\/"(X, T)
proof
  let T be adj-structured with_suprema antisymmetric commutative non empty
  non void reflexive transitive Noetherian TAS-structure;
  let t be type 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 = t};
A1: R is with_Church-Rosser_property with_UN_property strongly-normalizing
  Relation by Th69,Th77;
A2: radix t is_>=_than AA
  proof
    let tt be type of T;
    assume tt in AA;
    then ex t9 being type of T st tt = t9 & ex A being finite Subset of the
    adjectives of T st A is_properly_applicable_to t9 & A ast t9 = t;
    then R reduces t, tt by Th72;
    then t, tt are_convertible_wrt R by REWRITE1:25;
    then nf(t, R) = nf(tt, R) by A1,REWRITE1:55;
    then nf(t, R) is_a_normal_form_of tt, R by A1,REWRITE1:54;
    then R reduces tt, nf(t,R);
    hence thesis by Th67;
  end;
  ex A being finite Subset of the adjectives of T st A
  is_properly_applicable_to radix t & A ast radix t = t by Th76,Th78;
  then radix t in AA;
  then for t9 being type of T st t9 is_>=_than AA holds radix t <= t9;
  hence thesis by A2,YELLOW_0:30;
end;
