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

theorem Th12:
  A=[.0,r.] & r >=0 implies
     arctan.r = Partial_Sums(Leibniz_Series_of r).n +
     integral( (-1)|^(n+1) (#) ( #Z (2*(n+1))/ ( #Z 0 + #Z 2)),A)
proof
  set Z0 = #Z0,Z2 = #Z2,rL = Leibniz_Series_of r;
  assume A=[.0,r.] & r >=0;
  then
A1: upper_bound A = r & lower_bound A = 0 by JORDAN5A:19;
  defpred P[Nat] means arctan.r = Partial_Sums(rL).$1+
  integral((-1)|^($1+1)  (#) ( #Z (2*($1+1)) / ( Z0 + Z2)),A);
A2:  P[0]
  proof
A3: integral( Z0 / ( Z0 + Z2),A) = arctan.r - arctan.0 by Th5,A1
                                  .= arctan.r by SIN_COS9:43;
A4: 2*0+1=1;
A5: ((-1)|^0) * ((1*(r |^ 1)) -(1/1*(0 |^ 1)))
          = ((-1) |^0) * (r|^1/ 1)
         .= rL.0 by A4,Def2;
    (-1)|^0 = 1 by NEWTON:4;
    then (-1)|^0  (#) ( Z0/(Z0+Z2)) = Z0/(Z0+Z2) by RFUNCT_1:21;
    then arctan.r = rL.0 + integral((-1)|^(0+1) (#)
      ( #Z (2*(0+1)) / ( #Z 0 + #Z 2)),A) by A3,A1,A4,Th6,A5;
    hence thesis by SERIES_1:def 1;
  end;
A6: P[i] implies P[i+1]
  proof
    set i1=i+1,i11=i1+1;
    assume
A7:   P[i];
A8: 0 |^ (2*i1+1) =0 by NEWTON:11,NAT_1:11;
    ((-1)|^i1) * (((1/(2*i1+1))*(r |^ (2*i1+1))) -((1/(2*i1+1))*0))
    = (-1)|^i1 * (r |^ (2*i1+1))/(2*i1+1)
    .= rL.i1 by Def2;
    then integral( (-1)|^i1 (#) ( #Z (2*i1) / ( Z0 + Z2)),A)=
      rL.i1 +
      integral( (-1)|^i11 (#) ( #Z (2*i11) / ( Z0 + Z2)),A) by A8,Th6,A1;
    then arctan.r = Partial_Sums(rL).i+rL.i1 +
      integral( (-1)|^i11 (#) ( #Z (2*i11) / ( Z0 + Z2)),A) by A7;
    hence thesis by SERIES_1:def 1;
  end;
  P[i] from NAT_1:sch 2(A2,A6);
  hence thesis;
end;
