
theorem Th72:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema non void TAS-structure for t being type of T, A
  being finite Subset of the adjectives of T st A is_properly_applicable_to t
  holds T@--> reduces A ast t, t
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TAS-structure;
  set R = T@-->;
  let t be type of T, A be finite Subset of the adjectives of T;
  assume A is_properly_applicable_to t;
  then consider A9 being Subset of the adjectives of T such that
  A9 c= A and
A1: A9 is_properly_applicable_to t and
A2: A ast t = A9 ast t and
A3: for C being Subset of the adjectives of T st C c= A9 & C
  is_properly_applicable_to t & A ast t = C ast t holds C = A9 by Th64;
  consider s being one-to-one FinSequence of the adjectives of T such that
A4: rng s = A9 and
A5: s is_properly_applicable_to t by A1,Th65;
  reconsider p = Rev apply(s, t) as RedSequence of R by A2,A3,A4,A5,Th71;
  take p;
  thus p.1 = apply(s, t).len apply(s, t) by FINSEQ_5:62
    .= s ast t by Def19
    .= A ast t by A2,A4,A5,Th56,Th57;
  thus p.len p = p.len apply(s, t) by FINSEQ_5:def 3
    .= apply(s, t).1 by FINSEQ_5:62
    .= t by Def19;
end;
