
theorem Th13:
  for X be ComplexNormSpace, seq be sequence of X st (for n be
  Nat holds seq.n = 0.X) holds seq is norm_summable
proof
  let X be ComplexNormSpace;
  let seq be sequence of X such that
A1: for n be Nat holds seq.n = 0.X;
  take 0;
  let p be Real such that
A2: 0<p;
  take 0;
  let m be Nat such that
  0<=m;
  |.(Partial_Sums ||.seq.||).m-0 .| = |.0-0 .| by A1,Th6
    .= 0 by ABSVALUE:def 1;
  hence thesis by A2;
end;
