
theorem FINSEQ165:
  for n be non zero Nat, f,g be FinSequence holds
    (f^g).(len f + n) = g.n
  proof
    let n be non zero Nat, f,g be FinSequence;
    per cases;
    suppose
      n in dom g;
      hence thesis by FINSEQ_1:def 7;
    end;
    suppose
      B1: not n in dom g; then
      n < 1 or len g < n by FINSEQ_3:25; then
      len f + len g < len f + n by NAT_1:14,XREAL_1:6; then
      len (f^g) < len f + n by FINSEQ_1:22; then
      not (len f + n) in dom (f^g) by FINSEQ_3:25; then
      (f^g).(len f + n) = 0 by FUNCT_1:def 2;
      hence thesis by B1, FUNCT_1:def 2;
    end;
  end;
