
theorem Th70:
  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 for i being Nat st 1 <= i & i <=
  len s holds [apply(s, t).(i+1), apply(s, t).i] in 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;
  let j be Nat;
  assume that
A4: 1 <= j and
A5: j <= len s;
A6: len apply(s, t) = len s+1 by Def19;
  j < len s+1 by A5,NAT_1:13;
  then j in dom apply(s, t) by A6,A4,FINSEQ_3:25;
  then apply(s, t).j in rng apply(s, t) by FUNCT_1:3;
  then reconsider tt = apply(s, t).j as type of T;
A7: j in dom s by A4,A5,FINSEQ_3:25;
  then s.j in rng s by FUNCT_1:3;
  then reconsider a = s.j as adjective of T;
A8: apply(s, t).(j+1) = a ast tt by A7,Def19;
A9: not a in adjs tt
  proof
    assume
A10: a in adjs tt;
    consider i being Nat such that
A11: j = 1+i by A4,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;
    s1 c= s by TREES_1:def 1;
    then consider s2 being FinSequence such that
A12: s = s1^s2 by TREES_1:1;
    reconsider s2 as FinSequence of the adjectives of T by A12,FINSEQ_1:36;
A13: len s = len s1+len s2 by A12,FINSEQ_1:22;
    then
A14: s1 is_properly_applicable_to t by A3,A12,Th60;
    reconsider s21 = s2|Seg 1 as FinSequence of the adjectives of T by
FINSEQ_1:18;
    i <= len s by A5,A11,NAT_1:13;
    then
A15: len s1 = i by FINSEQ_1:17;
    then
A16: len s2 >= 1 by A5,A11,A13,XREAL_1:6;
    then
A17: len s21 = 1 by FINSEQ_1:17;
    then
A18: s21 = <*s21.1*> by FINSEQ_1:40;
    then
A19: rng s21 = {s21.1} by FINSEQ_1:39;
    then reconsider b = s21.1 as adjective of T by ZFMISC_1:31;
A20: 1 in dom s2 by A16,FINSEQ_3:25;
    s21 c= s2 by TREES_1:def 1;
    then consider s22 being FinSequence such that
A21: s2 = s21^s22 by TREES_1:1;
    reconsider s22 as FinSequence of the adjectives of T by A21,FINSEQ_1:36;
A22: rng s2 = rng s21 \/ rng s22 by A21,FINSEQ_1:31;
    then
A23: rng s22 c= rng s2 by XBOOLE_1:7;
A24: b = s2.1 by A18,A21,FINSEQ_1:41
      .= a by A11,A12,A15,A20,FINSEQ_1:def 7;
    then a in rng s21 by A19,TARSKI:def 1;
    then
A25: a in rng s2 by A22,XBOOLE_0:def 3;
    s1 ast t = tt by A11,A12,A13,A15,Th36;
    then
A26: s1 ast t = a ast (s1 ast t) by A10,Th24
      .= s21 ast (s1 ast t) by A18,A24,Th31;
    s2 is_properly_applicable_to s1 ast t by A3,A12,Th60;
    then s22 is_properly_applicable_to s1 ast t by A21,A26,Th60;
    then
A27: s1^s22 is_properly_applicable_to t by A14,Th61;
    reconsider B = rng (s1^s22) as Subset of the adjectives of T;
A28: B = rng s1 \/ rng s22 by FINSEQ_1:31;
A29: A = rng s1 \/ rng s2 by A2,A12,FINSEQ_1:31;
    s ast t = s2 ast (s1 ast t) by A12,Th37
      .= s22 ast (s1 ast t) by A21,A26,Th37
      .= s1^s22 ast t by Th37;
    then
A30: A ast t = s1^s22 ast t by A2,A3,Th56,Th57
      .= B ast t by A27,Th56,Th57;
    B is_properly_applicable_to t by A27;
    then B = A by A1,A30,A28,A29,A23,XBOOLE_1:9;
    then
A31: a in B by A29,A25,XBOOLE_0:def 3;
    per cases by A28,A31,XBOOLE_0:def 3;
    suppose
      a in rng s1;
      then consider x being object such that
A32:  x in dom s1 and
A33:  a = s1.x by FUNCT_1:def 3;
      reconsider x as Element of NAT by A32;
      x <= len s1 by A32,FINSEQ_3:25;
      then
A34:  x < j by A11,A15,NAT_1:13;
A35:  dom s1 c= dom s by A12,FINSEQ_1:26;
      s.x = a by A12,A32,A33,FINSEQ_1:def 7;
      hence contradiction by A7,A32,A35,A34,FUNCT_1:def 4;
    end;
    suppose
      a in rng s22;
      then consider x being object such that
A36:  x in dom s22 and
A37:  a = s22.x by FUNCT_1:def 3;
      reconsider x as Element of NAT by A36;
A38:  1+x in dom s2 by A17,A21,A36,FINSEQ_1:28;
      x >= 0+1 by A36,FINSEQ_3:25;
      then
A39:  j+x > j+0 by XREAL_1:6;
      s2.(1+x) = a by A17,A21,A36,A37,FINSEQ_1:def 7;
      then
A40:  s.(i+(1+x)) = a by A12,A15,A38,FINSEQ_1:def 7;
      i+(1+x) in dom s by A12,A15,A38,FINSEQ_1:28;
      hence contradiction by A7,A11,A39,A40,FUNCT_1:def 4;
    end;
  end;
  a is_properly_applicable_to tt by A3,A7;
  hence thesis by A8,A9,Def31;
end;
