
theorem Th6:
  for X be RealNormSpace for seq be sequence of X st (for n be
  Nat holds seq.n = 0.X)
   for m be Nat holds (Partial_Sums ||.seq.||).m = 0
proof
  let X be RealNormSpace;
  let seq be sequence of X such that
A1: for n be Nat holds seq.n = 0.X;
  let m be Nat;
  defpred P[Nat] means
||.seq.||.$1 = (Partial_Sums ||.seq.||).$1;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
    thus ||.seq.||.(k+1) = ||.0.X.|| + ||.seq.||.(k+1)
      .= ||.seq.k.|| + ||.seq.||.(k+1) by A1
      .= (Partial_Sums ||.seq.||).k + ||.seq.||.(k+1) by A3,NORMSP_0:def 4
      .= (Partial_Sums ||.seq.||).(k+1) by SERIES_1:def 1;
  end;
A4: P[0] by SERIES_1:def 1;
  for n be Nat holds P[n] from NAT_1:sch 2(A4,A2);
  hence (Partial_Sums ||.seq.||).m = ||.seq.||.m
    .= ||.seq.m.|| by NORMSP_0:def 4
    .= ||.0.X.|| by A1
    .= 0;
end;
