theorem
  F[:](p^q,d9) = (F[:](p,d9))^(F[:](q,d9))
proof
  defpred P[FinSequence of D] means F[:](p^($1),d9) = (F[:](p,d9))^(F[:]($1,d9
  ));
A1: for q,d st P[q] holds P[q^<*d*>]
  proof
    let q,d such that
A2: F[:](p^q,d9) = (F[:](p,d9))^(F[:](q,d9));
    thus F[:](p^(q^<*d*>),d9) = F[:]((p^q)^<*d*>,d9) by FINSEQ_1:32
      .= (F[:](p^q,d9))^<*F.(d,d9)*> by Th14
      .= (F[:](p,d9))^((F[:](q,d9))^<*F.(d,d9)*>) by A2,FINSEQ_1:32
      .= (F[:](p,d9))^(F[:](q^<*d*>,d9)) by Th14;
  end;
  F[:](p^(<*>D),d9) = F[:](p,d9) by FINSEQ_1:34
    .= (F[:](p,d9))^(<*>E) by FINSEQ_1:34
    .= (F[:](p,d9))^(F[:](<*>D,d9)) by FINSEQ_2:85;
  then
A3: P[<*>D];
  for q holds P[q] from FINSEQ_2:sch 2(A3,A1);
  hence thesis;
end;
