reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th10:
for seq be Real_Sequence, n,m be Nat st m <= n holds
  |.(Partial_Sums seq).n - (Partial_Sums seq).m.|
   <= (Partial_Sums abs seq).n - (Partial_Sums abs seq).m &
  |.(Partial_Sums seq).n - (Partial_Sums seq).m.|
   <= (Partial_Sums abs seq).n
proof
   let seq be Real_Sequence;
   let n, m be Nat;
   assume A1: m<= n;
A2:for n holds abs(seq).n >= 0
   proof
    let n;
    |.seq.n.|=abs(seq).n by SEQ_1:12;
    hence thesis by COMPLEX1:46;
   end; then
A3: |.(Partial_Sums abs seq).n - (Partial_Sums abs seq).m.|
    = (Partial_Sums abs seq).n - (Partial_Sums abs seq).m by A1,COMSEQ_3:9;
   (Partial_Sums abs seq).m >= 0 by A2,SERIES_3:34; then
   |.(Partial_Sums seq).n - (Partial_Sums seq).m.|
    <= (Partial_Sums abs seq).n - (Partial_Sums abs seq).m
      + (Partial_Sums abs seq).m by A3,A1,SERIES_1:34,XREAL_1:38;
   hence thesis by A3,A1,SERIES_1:34;
end;
