
theorem Th71:
  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 for C being Subset of the
  adjectives of T st C c= A & C is_properly_applicable_to t & A ast t = C ast t
holds C = A for s being one-to-one FinSequence of the adjectives of T st rng s
  = A & s is_properly_applicable_to t holds Rev apply(s, t) is RedSequence of T
  @-->
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TAS-structure;
  let t be type of T, A be finite Subset of the adjectives of T such that
A1: for C being Subset of the adjectives of T st C c= A & C
  is_properly_applicable_to t & A ast t = C ast t holds C = A;
  let s be one-to-one FinSequence of the adjectives of T such that
A2: rng s = A and
A3: s is_properly_applicable_to t;
A4: len Rev apply(s, t) = len apply(s, t) by FINSEQ_5:def 3;
  hence len Rev apply(s, t) > 0;
  let i be Nat;
  assume that
A5: i in dom Rev apply(s, t) and
A6: i+1 in dom Rev apply(s, t);
A7: len apply(s, t) = len s+1 by Def19;
  then
A8: (Rev apply(s, t)).i = apply(s, t).(len s+1 -i +1) by A5,FINSEQ_5:def 3;
  i+1 <= len s+1 by A4,A7,A6,FINSEQ_3:25;
  then consider j being Nat such that
A9: len s+1 = i+1+j by NAT_1:10;
A10: (Rev apply(s, t)).(i+1) = apply(s, t).(len s+1-(i+1)+1) by A7,A6,
FINSEQ_5:def 3;
A11: i >= 1 by A5,FINSEQ_3:25;
  len s = i+j by A9;
  then
A12: j+1 <= len s by A11,XREAL_1:6;
  j+1 >= 1 by NAT_1:11;
  hence thesis by A1,A2,A3,A8,A10,A9,A12,Th70;
end;
