theorem Th51:
  (for n holds seq.n = 0c) implies seq is absolutely_summable
proof
  assume
A1: for n holds seq.n = 0c;
  take 0;
  let p be Real such that
A2: 0<p;
  take 0;
  let m;
  |.(Partial_Sums |.seq.|).m-0.| = |.0-0.| by A1,Th25
    .= 0 by ABSVALUE:def 1;
  hence thesis by A2;
end;
