
theorem Th37:
  for T being non empty non void reflexive transitive TA-structure
for t being type of T for v1,v2 being FinSequence of the adjectives of T holds
  v2 ast (v1 ast t) = v1^v2 ast t
proof
  let T be non empty non void reflexive transitive TA-structure;
  let t be type of T;
  let v1,v2 be FinSequence of the adjectives of T;
  set v = v1^v2;
  consider tt being FinSequence of the carrier of T, q being Element of T such
  that
A1: apply(v1,t) = tt^<*q*> by HILBERT2:4;
A2: len apply(v1,t) = len v1+1 by Def19;
  len <*q*> = 1 by FINSEQ_1:39;
  then
A3: len v1+1 = len tt+1 by A2,A1,FINSEQ_1:22;
A4: len v2+1 >= 1 by NAT_1:11;
  len apply(v2, v1 ast t) = len v2+1 by Def19;
  then
A5: len v2+1 in dom apply(v2, v1 ast t) by A4,FINSEQ_3:25;
  apply(v,t) = apply(v1,t) $^ apply(v2, v1 ast t) by Th34
    .= tt^apply(v2, v1 ast t) by A1,REWRITE1:2;
  hence v2 ast (v1 ast t) = apply(v,t).(len tt+(len v2+1)) by A5,FINSEQ_1:def 7
    .= apply(v,t).(len v1+len v2+1) by A3
    .= v ast t by FINSEQ_1:22;
end;
