
theorem Th46:
  for T being Noetherian adj-structured reflexive transitive
antisymmetric with_suprema non void TA-structure for t being type of T for v1,
  v2 being FinSequence of the adjectives of T st v1 is_applicable_to t & rng v2
  c= rng v1 for s being type of T st s in rng apply(v2,t) holds v1 ast t <= s
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TA-structure;
  let t be type of T;
  let v,v9 be FinSequence of the adjectives of T such that
A1: v is_applicable_to t and
A2: rng v9 c= rng v;
  defpred P[Nat] means $1 <= len apply(v9,t) implies for s being type of T st
  s = apply(v9,t).$1 holds v ast t <= s;
A3: for i being non zero Nat st P[i] holds P[i+1]
  proof
A4: rng v c= adjs (v ast t) by A1,Th44;
    let i be non zero Nat such that
A5: P[i] and
A6: i+1 <= len apply(v9,t);
A7: 0+1 <= i by NAT_1:13;
A8: len apply(v9,t) = len v9+1 by Def19;
    then i < len v9+1 by A6,NAT_1:13;
    then i in dom apply(v9,t) by A8,A7,FINSEQ_3:25;
    then apply(v9,t).i in rng apply(v9,t) by FUNCT_1:3;
    then reconsider s = apply(v9,t).i as type of T;
A9: v ast t <= s by A5,A6,NAT_1:13;
    i <= len v9 by A6,A8,XREAL_1:6;
    then
A10: i in dom v9 by A7,FINSEQ_3:25;
    then
A11: v9.i in rng v9 by FUNCT_1:3;
    then reconsider a = v9.i as adjective of T;
A12: a in rng v by A2,A11;
    apply(v9,t).(i+1) = a ast s by A10,Def19;
    hence thesis by A12,A4,A9,Th23;
  end;
  apply(v9,t).1 = t by Def19;
  then
A13: P[1] by A1,Th43;
A14: for i being non zero Nat holds P[i] from NAT_1:sch 10(A13,A3);
  let s be type of T;
  assume s in rng apply(v9,t);
  then consider x being object such that
A15: x in dom apply(v9,t) and
A16: s = apply(v9,t).x by FUNCT_1:def 3;
A17: x in Seg len apply(v9,t) by A15,FINSEQ_1:def 3;
  reconsider x as Element of NAT by A15;
  reconsider x as non zero Element of NAT by A17,FINSEQ_1:1;
  x <= len apply(v9,t) by A15,FINSEQ_3:25;
  hence thesis by A16,A14;
end;
