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

theorem Th9:
  r in [.-1,1.] implies abs(Leibniz_Series_of r) is
        nonnegative-yielding non-increasing convergent
     & lim abs (Leibniz_Series_of r) = 0
proof
  set rL=Leibniz_Series_of r,A=abs rL;
  assume
A1:r in [.-1,1.];
A2:dom A = dom rL & dom rL=NAT by VALUED_1:def 11,FUNCT_2:def 1;
  thus A is  nonnegative-yielding;
A3: A.n = |.r.| |^ (2*n+1)/ (2*n+1)
  proof
    --1=1;
    then |.-1.| = 1 by ABSVALUE:def 1;
    then
A4: |.(-1)|^n.| = 1 |^n by TAYLOR_2:1;
A5: A.n = |.rL.n.| by A2,VALUED_1:def 11,ORDINAL1:def 12;
    rL.n = (-1)|^n * (r|^(2*n+1))/ (2*n+1) by Def2;
    hence  A.n = |.(-1)|^n *(r|^(2*n+1)).|/ |.(2*n+1).| by COMPLEX1:67,A5
              .= 1*|.(r|^(2*n+1)).|/ |.(2*n+1).| by A4,COMPLEX1:65
              .= 1*(|.r.| |^(2*n+1)) / |.(2*n+1).| by TAYLOR_2:1
              .= |.r.| |^(2*n+1) / (2*n+1) by ABSVALUE:def 1;
  end;
  -1 <=r <= 1 by A1,XXREAL_1:1;
  then
A6: |.r.| <= 1 by ABSVALUE:5;
  A.n >= A.(n+1)
  proof
    set n1=n+1;
A7: A.n1 = |.r.| |^ (2*n1+1)*1/(2*n1+1) by A3;
A8: |.r.| >=0 by COMPLEX1:46;
    |.(r|^(2*n+1)).| = |.r.||^(2*n+1) by TAYLOR_2:1;
    then
A9: |.r.| |^(2*n+1)>=0 by COMPLEX1:46;
A10: |.r.| * |.r.| <=1*1 by A8,A6,XREAL_1:66;
    |.r.| |^ (2*n1+1) = |.r.| * |.r.| |^ (2*n+1+1) by NEWTON:6
                     .= |.r.| *( |.r.| * |.r.| |^(2*n+1)) by NEWTON:6
                     .= (|.r.|*|.r.|) * (|.r.| |^(2*n+1));
    then
A11: |.r.| |^ (2*n1+1) <= |.r.| |^(2*n+1) *1 by A9,A10,XREAL_1:66;
    |.(r|^(2*n1+1)).| = |.r.||^(2*n1+1) by TAYLOR_2:1;
    then
A12: |.r.| |^ (2*n1+1)>=0 by COMPLEX1:46;
    2*n+1 <=2*n+1+1 by NAT_1:13;
    then 2*n+1 < 2*n+1+1+1 by NAT_1:13;
    then 1/(2*n+1) >= 1/(2*n1+1) by XREAL_1:76;
    then |.r.| |^ (2*n+1)*1/(2*n+1) >= |.r.| |^ (2*n1+1) *1/(2*n1+1)
      by A12,A11,XREAL_1:66;
    hence thesis by A3,A7;
  end;
  hence A is non-increasing by VALUED_1:def 16;
  set C=seq_const 0;
A13: lim C =0;
  deffunc F(Nat) = (1/2) / ($1 +1/2);
  consider f be Real_Sequence such that
A14: f.n=F(n) from SEQ_1:sch 1;
A15:f is convergent & lim f =0 by A14,SEQ_4:31;
  C.n<=A.n<=f.n
  proof
A17:  |.r.| >=0 by COMPLEX1:46;
A18:   0 |^(2*n+1)=0 by NEWTON:11,NAT_1:11;
    |.r.| >0 implies
    |.r.| |^ (2*n+1) <= 1 |^ (2*n+1) by A6,PREPOWER:9;
    then
A19: |.r.| |^ (2*n+1) <= 1 by A18,A17;
    |.(r|^(2*n+1)).| = |.r.||^(2*n+1) by TAYLOR_2:1;
    then
A20: |.r.| |^(2*n+1)>=0 by COMPLEX1:46;
    (2*n+1)/2=n+1/2;
    then
A21:  1/(2*n+1) = F(n) by XCMPLX_1:55
               .=f.n by A14;
    A.n = |.r.| |^ (2*n+1)/ (2*n+1) by A3
       .= |.r.| |^ (2*n+1)*(2*n+1)";
    hence thesis by A19, XREAL_1:64,A20,A21;
  end;
  hence thesis by A13,A15,SEQ_2:19,20;
end;
