
theorem Th65:
  for T being Noetherian adj-structured reflexive transitive
  antisymmetric with_suprema non void TAS-structure for t being type of T, A
being Subset of the adjectives of T st A is_properly_applicable_to t ex s being
  one-to-one FinSequence of the adjectives of T st rng s = A & s
  is_properly_applicable_to 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 Subset of the adjectives of T;
  given s9 being FinSequence of the adjectives of T such that
A1: rng s9 = A and
A2: s9 is_properly_applicable_to t;
  defpred P[Nat] means ex s being FinSequence of the adjectives of T st $1 =
  len s & rng s = A & s is_properly_applicable_to t;
  len s9 = len s9;
  then
A3: ex k being Nat st P[k] by A1,A2;
  consider k being Nat such that
A4: P[k] and
A5: for n being Nat st P[n] holds k <= n from NAT_1:sch 5(A3);
  consider s being FinSequence of the adjectives of T such that
A6: k = len s and
A7: rng s = A and
A8: s is_properly_applicable_to t by A4;
  s is one-to-one
  proof
    let x,y be object;
    assume that
A9: x in dom s and
A10: y in dom s and
A11: s.x = s.y and
A12: x <> y;
    reconsider x,y as Element of NAT by A9,A10;
    x < y or x > y by A12,XXREAL_0:1;
    then consider x,y being Element of NAT such that
A13: x in dom s and
A14: y in dom s and
A15: x < y and
A16: s.x = s.y by A9,A10,A11;
A17: x >= 1 by A13,FINSEQ_3:25;
    y >= 1 by A14,FINSEQ_3:25;
    then consider i being Nat such that
A18: y = 1+i by NAT_1:10;
    reconsider i as Element of NAT by ORDINAL1:def 12;
    reconsider s1 = s|Seg i as FinSequence of the adjectives of T by
FINSEQ_1:18;
A19: y <= len s by A14,FINSEQ_3:25;
    then i <= len s by A18,NAT_1:13;
    then
A20: len s1 = i by FINSEQ_1:17;
    x <= i by A15,A18,NAT_1:13;
    then
A21: x in dom s1 by A20,A17,FINSEQ_3:25;
    s1 c= s by TREES_1:def 1;
    then consider s2 being FinSequence such that
A22: s = s1^s2 by TREES_1:1;
    reconsider s2 as FinSequence of the adjectives of T by A22,FINSEQ_1:36;
A23: len s = len s1+len s2 by A22,FINSEQ_1:22;
    then
A24: len s2 >= 1 by A18,A19,A20,XREAL_1:6;
    then
A25: 1 in dom s2 by FINSEQ_3:25;
    reconsider s21 = s2|Seg 1 as FinSequence of the adjectives of T by
FINSEQ_1:18;
    s21 c= s2 by TREES_1:def 1;
    then consider s22 being FinSequence such that
A26: s2 = s21^s22 by TREES_1:1;
    reconsider s22 as FinSequence of the adjectives of T by A26,FINSEQ_1:36;
A27: len s21 = 1 by A24,FINSEQ_1:17;
    then
A28: s21 = <*s21.1*> by FINSEQ_1:40;
    then
A29: rng s21 = {s21.1} by FINSEQ_1:39;
    then reconsider a = s21.1 as adjective of T by ZFMISC_1:31;
A30: rng s2 = rng s21 \/ rng s22 by A26,FINSEQ_1:31;
    a = s2.1 by A28,A26,FINSEQ_1:41
      .= s.y by A18,A22,A20,A25,FINSEQ_1:def 7;
    then a = s1.x by A16,A22,A21,FINSEQ_1:def 7;
    then
A31: a in rng s1 by A21,FUNCT_1:3;
    then rng s21 c= rng s1 by A29,ZFMISC_1:31;
    then rng s1 \/ rng s21 = rng s1 by XBOOLE_1:12;
    then
A32: rng (s1^s22) = rng s1 \/ rng s21 \/ rng s22 by FINSEQ_1:31;
A33: s2 is_properly_applicable_to s1 ast t by A8,A22,Th60;
A34: s1 is_properly_applicable_to t by A8,A22,A23,Th60;
    then rng s1 c= adjs (s1 ast t) by Th44,Th57;
    then s1 ast t = a ast (s1 ast t) by A31,Th24
      .= s21 ast (s1 ast t) by A28,Th31;
    then s22 is_properly_applicable_to s1 ast t by A26,A33,Th60;
    then
A35: s1^s22 is_properly_applicable_to t by A34,Th61;
A36: len s2 = len s21+len s22 by A26,FINSEQ_1:22;
    rng s = rng s1 \/ rng s2 by A22,FINSEQ_1:31;
    then k <= len (s1^s22) by A5,A7,A35,A32,A30,XBOOLE_1:4;
    then k <= len s1+len s22 by FINSEQ_1:22;
    then len s21+len s22 <= 0+len s22 by A6,A23,A36,XREAL_1:6;
    hence thesis by A27,XREAL_1:6;
  end;
  hence thesis by A7,A8;
end;
