reserve n for Nat,
  a,b for Real,
  s for Real_Sequence;

theorem
  (for n holds s.n =((-1)|^n*2|^(n+1)) /((2|^(n+1)+(-1)|^(n+1))*(2|^(n+2
)+(-1)|^(n+2)))) implies for n holds Partial_Sums(s).n = 1/3+((-1)|^(n+2))/(3*(
  2|^(n+2)+(-1)|^(n+2)))
proof
  defpred X[Nat] means Partial_Sums(s).$1 = 1/3+((-1)|^($1+2))/(3*(
  2|^($1+2)+(-1)|^($1+2)));
  assume
A1: for n holds s.n =((-1)|^n*2|^(n+1)) /((2|^(n+1)+(-1)|^(n+1))*(2|^(n+
  2)+(-1)|^(n+2)));
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    set b=2|^(n+2)+(-1)|^(n+2);
    set p=2|^(n+3)+(-1)|^(n+3);
A3: b>0 by Lm18;
    assume Partial_Sums(s).n =1/3+((-1)|^(n+2))/(3*(2|^(n+2)+(-1)|^(n+2)));
    then
A4: Partial_Sums(s).(n+1) =1/3+((-1)|^(n+2))/(3*(2|^(n+2)+(-1)|^(n+2)))+s.
    (n+1) by SERIES_1:def 1
      .=1/3+((-1)|^(n+2))/(3*(2|^(n+2)+(-1)|^(n+2))) +((-1)|^(n+1)*2|^((n+1)
    +1))/((2|^((n+1)+1)+(-1)|^((n+1)+1)) *(2|^((n+1)+2)+(-1)|^((n+1)+2))) by A1
      .=1/3+((-1)|^(n+2))/(3*(2|^(n+2)+(-1)|^(n+2)))+((-1)|^(n+1)*2|^(n+2))
    /((2|^(n+2)+(-1)|^(n+2))*(2|^(n+3)+(-1)|^(n+3)));
    p>0 by Lm19;
    then
    Partial_Sums(s).(n+1) =1/3+((-1)|^(n+2)*p)/(3*b*p)+((-1)|^(n+1)*2|^(n+
    2))/(b*p) by A4,XCMPLX_1:91
      .=1/3+((-1)|^(n+2)*p)/(3*b*p)+((-1)|^(n+1)*2|^(n+2)*3)/(b*p*3) by
XCMPLX_1:91
      .=1/3+(((-1)|^(n+2)*p)/(3*b*p)+((-1)|^(n+1)*2|^(n+2)*3)/(3*b*p))
      .=1/3+(((-1)|^(n+2)*(-1)/(-1)*p)+((-1)|^(n+1)*2|^(n+2)*3))/(3*b*p) by
XCMPLX_1:62
      .=1/3+(((-1)|^(n+2+1)/(-1)*p)+((-1)|^(n+1)*2|^(n+2)*3))/(3*b*p) by
NEWTON:6
      .=1/3+(((-1)|^(n+3)/(-1)*p)+((-1)|^(n+1)*(-1)/(-1)*2|^(n+2)*3)) /(3*b*
    p)
      .=1/3+(((-1)|^(n+3)/(-1)*p)+((-1)|^(n+1+1)/(-1)*2|^(n+2)*3)) /(3*b*p)
    by NEWTON:6
      .=1/3+(((-1)|^(n+3)/(-1)*p)+((-1)|^(n+2)*(-1)/((-1)*(-1))*2|^(n+2)*3))
    /(3*b*p)
      .=1/3+(((-1)|^(n+3)/(-1)*p)+((-1)|^(n+2+1)/((-1)*(-1))*2|^(n+2)*3)) /(
    3*b*p) by NEWTON:6
      .=1/3+((-1)|^(n+3)*(2|^(n+2)+2|^(n+2)*2-2|^(n+3)-(-1)|^(n+3)))/(3*b*p)
      .=1/3+((-1)|^(n+3)*(2|^(n+2)+2|^(n+2+1)-2|^(n+3)-(-1)|^(n+3)))/(3*b*p)
    by NEWTON:6
      .=1/3+((-1)|^(n+3)*(2|^(n+2)-(-1)|^(n+2)*(-1)))/(3*b*p) by NEWTON:6
      .=1/3+((-1)|^(n+3)*b)/(3*p*b)
      .=1/3+((-1)|^(n+2+1))/(3*(2|^(n+2+1)+(-1)|^(n+2+1))) by A3,XCMPLX_1:91;
    hence thesis;
  end;
  Partial_Sums(s).0 = s.0 by SERIES_1:def 1
    .=((-1)|^0*2|^(0+1)) /((2|^(0+1)+(-1)|^(0+1))*(2|^(0+2)+(-1)|^(0+2))) by A1
    .=(1*2|^(0+1))/((2|^(0+1)+(-1)|^(0+1))*(2|^(0+2)+(-1)|^(0+2))) by NEWTON:4
    .=(1*2)/((2|^(0+1)+(-1)|^(0+1))*(2|^(0+2)+(-1)|^(0+2)))
    .=(1*2)/((2+(-1)|^(0+1))*(2|^(0+2)+(-1)|^(0+2)))
    .=(1*2)/((2+(-1))*(2|^(0+2)+(-1)|^(0+2)))
    .=(1*2)/((2+(-1))*(2*2+(-1)|^(0+2))) by WSIERP_1:1
    .=(1*2)/((2+(-1))*(2*2+(-1)*(-1))) by WSIERP_1:1
    .=1/3+((-1)*(-1))/(3*(2*2+(-1)*(-1)))
    .=1/3+((-1)|^2)/(3*(2*2+(-1)*(-1))) by WSIERP_1:1
    .=1/3+((-1)|^2)/(3*(2*2+(-1)|^2)) by WSIERP_1:1
    .=1/3+((-1)|^(0+2))/(3*(2|^(0+2)+(-1)|^(0+2))) by WSIERP_1:1;
  then
A5: X[0];
  for n holds X[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
