
theorem Th47:
  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^v2 is_applicable_to t holds
  v2^v1 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 v1,v2 be FinSequence of the adjectives of T;
A1: rng (v1^v2) = rng v1 \/ rng v2 by FINSEQ_1:31;
  assume
A2: v1^v2 is_applicable_to t;
  then
A3: rng (v1^v2) c= adjs ((v1^v2) ast t) by Th44;
  let i be Nat, a be adjective of T, s be type of T such that
A4: i in dom (v2^v1) and
A5: a = (v2^v1).i and
A6: s = apply(v2^v1,t).i;
A7: a in rng (v2^v1) by A4,A5,FUNCT_1:3;
A8: len apply(v2^v1,t) = len (v2^v1)+1 by Def19;
A9: rng (v2^v1) = rng v1 \/ rng v2 by FINSEQ_1:31;
  i <= len (v2^v1) by A4,FINSEQ_3:25;
  then
A10: i < len (v2^v1)+1 by NAT_1:13;
  i >= 1 by A4,FINSEQ_3:25;
  then i in dom apply(v2^v1,t) by A10,A8,FINSEQ_3:25;
  then s in rng apply(v2^v1,t) by A6,FUNCT_1:3;
  then (v1^v2) ast t <= s by A2,A1,A9,Th46;
  hence thesis by A1,A9,A7,A3,Th23;
end;
