
theorem Th35:
  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 for i
  being Nat st i in dom v1 holds apply(v1^v2, t).i = apply(v1, t).i
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;
  let i be Nat;
A2: len apply(v1,t) = len v1+1 by Def19;
  assume
A3: i in dom v1;
  then
A4: i >= 1 by FINSEQ_3:25;
  len <*q*> = 1 by FINSEQ_1:39;
  then len v1+1 = len tt+1 by A2,A1,FINSEQ_1:22;
  then i <= len tt by A3,FINSEQ_3:25;
  then
A5: i in dom tt 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;
  then apply(v,t).i = tt.i by A5,FINSEQ_1:def 7;
  hence thesis by A1,A5,FINSEQ_1:def 7;
end;
