
theorem
  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
  v1^v2 ast t = v2^v1 ast 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;
  assume
A1: v1^v2 is_applicable_to t;
A2: len (v1^v2) = len v1+len v2 by FINSEQ_1:22;
A3: rng (v1^v2) = rng v1 \/ rng v2 by FINSEQ_1:31;
A4: len (v2^v1) = len v1+len v2 by FINSEQ_1:22;
A5: len (v1^v2)+1 >= 1 by NAT_1:11;
A6: rng (v2^v1) = rng v1 \/ rng v2 by FINSEQ_1:31;
  len apply(v2^v1, t) = len (v2^v1)+1 by Def19;
  then len (v1^v2)+1 in dom apply(v2^ v1, t) by A2,A4,A5,FINSEQ_3:25;
  then v2^v1 ast t in rng apply(v2^v1, t) by A2,A4,FUNCT_1:3;
  then
A7: v1^v2 ast t <= v2^v1 ast t by A1,A3,A6,Th46;
  len apply(v1^v2, t) = len (v1^v2)+1 by Def19;
  then len (v1^v2)+1 in dom apply(v1^v2, t) by A5,FINSEQ_3:25;
  then v1^v2 ast t in rng apply(v1^v2, t) by FUNCT_1:3;
  then v2^v1 ast t <= v1^v2 ast t by A1,A3,A6,Th46,Th47;
  hence thesis by A7,YELLOW_0:def 3;
end;
