reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem Th41:
  for n holds |.(Partial_Sums s).n.|<=(Partial_Sums (abs s)).n
proof
  set s1=abs s;
  defpred X[Nat] means |.(Partial_Sums s).$1.|<=(Partial_Sums s1).
  $1;
  let n;
A1: for n st X[n] holds X[n+1]
  proof
    let n;
    assume |.(Partial_Sums(s)).n.|<=(Partial_Sums(s1)).n;
    then
A2: |.(Partial_Sums(s)).n.|+|.s.(n+1).|<=(Partial_Sums(s1)).n + |.s.(
    n+1).| by XREAL_1:6;
    (Partial_Sums(s1)).(n+1)=(Partial_Sums(s1)).n + s1.(n+1) by SERIES_1:def 1;
    then
A3: (Partial_Sums(s1)).(n+1)=(Partial_Sums(s1)).n + |.s.(n+1).| by SEQ_1:12;
    |.(Partial_Sums(s)).(n+1).|=|.(Partial_Sums(s)).n + s.(n+1).| & |.
    ( Partial_Sums(s)).n + s.(n+1).|<=|.(Partial_Sums(s)).n.| + |.s.(n+1).|
by COMPLEX1:56,SERIES_1:def 1;
    hence thesis by A3,A2,XXREAL_0:2;
  end;
  s1.0=|.s.0.| by SEQ_1:12;
  then (Partial_Sums s1).0=|.s.0.| by SERIES_1:def 1;
  then
A4: X[0] by SERIES_1:def 1;
  for n holds X[n] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
