
theorem Th42:
  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 s being
  type of T st s in rng apply(v, t) holds v ast t <= s & s <= t
proof
  let T be Noetherian adj-structured reflexive transitive antisymmetric
  with_suprema non void TA-structure;
  let t be type of T;
  let v be FinSequence of the adjectives of T such that
A1: v is_applicable_to t;
A2: len apply(v,t) = len v+1 by Def19;
  let s be type of T;
  assume s in rng apply(v,t);
  then consider x being object such that
A3: x in dom apply(v,t) and
A4: s = apply(v,t).x by FUNCT_1:def 3;
  reconsider x as Element of NAT by A3;
A5: x <= len apply(v,t) by A3,FINSEQ_3:25;
A6: apply(v,t).1 = t by Def19;
  x >= 1 by A3,FINSEQ_3:25;
  hence thesis by A1,A4,A5,A2,A6,Th41;
end;
