
theorem Th43:
  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 holds v ast t
  <= 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 v+1 >= 1 by NAT_1:11;
  len apply(v,t) = len v+1 by Def19;
  then len v+1 in dom apply(v, t) by A2,FINSEQ_3:25;
  then apply(v,t).(len v+1) in rng apply(v,t) by FUNCT_1:3;
  hence thesis by A1,Th42;
end;
