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

theorem Th11:
  r in [.-1,1.] implies Leibniz_Series_of r is summable
  proof
    set rL=Leibniz_Series_of r,A=abs rL,aA=alternating_series A;
    assume r in [.-1,1.];
    then A is nonnegative-yielding non-increasing convergent
      & lim A = 0 by Th9;
    then
A1: aA is summable by Th8;
    per cases;
    suppose r >=0;
      hence thesis by Th10,A1;
    end;
    suppose r <0;
      then rL = (-1)(#)aA by Th10;
      hence thesis by A1,SERIES_1:10;
    end;
  end;
