
theorem PPM:
  for f be Complex_Sequence, n,m be Nat st f.n = 0 holds
    (Partial_Product f).(n+m) = 0
  proof
    let f be Complex_Sequence, n,m be Nat such that
    A1: f.n = 0;
    defpred P[Nat] means (Partial_Product f).(n+$1) = 0;
    A2: P[0] by A1,PPN;
    A3: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat such that
      B1: P[k];
      (Partial_Product f).(n+k+1) = (Partial_Product f).(n+k)* f.(n+k+1) by PP;
      hence thesis by B1;
    end;
    for x be Nat holds P[x] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
