
theorem FINSEQ1D7:
  for f be FinSequence, n be Nat holds (f^<*0*>).n = f.n
  proof
    let f be FinSequence, n be Nat;
    reconsider g = <*0*> as FinSequence;
    per cases;
    suppose n in dom f;
      hence thesis by FINSEQ_1:def 7;
    end;
    suppose
      not n in dom f; then
      n < 1 or len f < n by FINSEQ_3:25; then
      per cases by NAT_1:14;
      suppose
        n = 0;
        hence thesis;
      end;
      suppose
        n > len f; then
        n - len f > len f - len f by XREAL_1:9; then
        reconsider m = n - len f as non zero Nat;
        (f^g).(len f + m) = g.m by FINSEQ165
        .= 0;
        hence thesis;
      end;
    end;
  end;
