reserve i,n,m for Nat,
        r,s for Real,
        A for non empty closed_interval Subset of REAL;

theorem Th15:
   Partial_Sums(Leibniz_Series).(2*n+1)
         <= Sum Leibniz_Series <=
   Partial_Sums(Leibniz_Series).(2*n)
proof
   set L=Leibniz_Series,A=abs L,aa = alternating_series A;
   1 in [.-1,1.] by XXREAL_1:1;
   then
A1:  A is nonnegative-yielding non-increasing convergent &
     lim A =0 by Th9;
   aa = L by Th10;
   hence thesis by A1,Th8;
end;
