theorem
  (Partial_Sums s).n<=(Partial_Sums abs s).n
proof
  set s1=abs s;
  (Partial_Sums s).n<=|.(Partial_Sums s).n.| & |.(Partial_Sums s).n.|<=(
  Partial_Sums s1).n by Th41,ABSVALUE:4;
  hence thesis by XXREAL_0:2;
end;
