
theorem Th56:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema non void TA-structure for t being type of T for v
  being FinSequence of the adjectives of T st v is_applicable_to t for A being
  Subset of the adjectives of T st A = rng v holds v ast t = A ast t
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TA-structure;
  defpred P[Nat] means for t being type of T, v being FinSequence
of the adjectives of T st $1 = len v & v is_applicable_to t for A being Subset
  of the adjectives of T st A = rng v holds v ast t = A ast t;
  let t be type of T;
  let v be FinSequence of the adjectives of T;
A1: now
    let n be Nat such that
A2: P[n];
    now
      let t be type of T, v be FinSequence of the adjectives of T such that
A3:   n+1 = len v and
A4:   v is_applicable_to t;
      consider v1 being FinSequence of the adjectives of T, a being Element of
      the adjectives of T such that
A5:   v = v1^<*a*> by A3,FINSEQ_2:19;
      reconsider B = rng v1 as Subset of the adjectives of T;
      reconsider a as adjective of T;
      len <*a*> = 1 by FINSEQ_1:40;
      then
A6:   len v = len v1+1 by A5,FINSEQ_1:22;
      v1 is_applicable_to t by A4,A5,Th40;
      then
A7:   v1 ast t = B ast t by A2,A3,A6;
      let A be Subset of the adjectives of T;
      assume
A8:   A = rng v;
      then
A9:   A = B \/ rng <*a*> by A5,FINSEQ_1:31
        .= B \/ {a} by FINSEQ_1:38;
      thus v ast t = <*a*> ast (v1 ast t) by A5,Th37
        .= a ast (B ast t) by A7,Th31
        .= A ast t by A4,A8,A9,Th45,Th55;
    end;
    hence P[n+1];
  end;
A10: P[0]
  proof
    let t be type of T;
    let v be FinSequence of the adjectives of T;
    assume
A11: 0 = len v;
    then v = <*> the adjectives of T;
    then
A12: rng v = {} the adjectives of T;
    v ast t = t by A11,Def19;
    hence thesis by A12,Th27;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A10,A1);
  hence thesis;
end;
