reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem
  (for n holds 0<=s.n) & s is summable implies s is absolutely_summable
proof
  assume that
A1: for n holds 0<=s.n and
A2: s is summable;
A3: for n holds s.n = |.s.n.| by A1,ABSVALUE:def 1;
  Partial_Sums(s) is convergent by A2;
  then Partial_Sums(abs(s)) is convergent by A3,SEQ_1:12;
  then abs(s) is summable;
  hence thesis;
end;
