
theorem Th45:
  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 A being
  Subset of the adjectives of T st A = rng v holds A is_applicable_to 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;
  assume
A1: v is_applicable_to t;
  then
A2: rng v c= adjs (v ast t) by Th44;
  v ast t <= t by A1,Th43;
  hence thesis by A2;
end;
