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

theorem
  (for n st n>=1 holds s.n = 1/((2*n-1)*(2*n+1)*(2*n+3)) & s.0=0)
  implies for n st n>=1 holds Partial_Sums(s).n =1/12-1/(4*(2*n+1)*(2*n+3))
proof
  defpred X[Nat] means Partial_Sums(s).$1= 1/12-1/(4*(2*$1+1)*(2*$1+3));
  assume
A1: for n st n>=1 holds s.n =1/((2*n-1)*(2*n+1)*(2*n+3)) & s.0=0;
A2: for n be Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
    n>=1 and
A3: Partial_Sums(s).n =1/12-1/(4*(2*n+1)*(2*n+3));
A4: n+1>=1 by NAT_1:11;
    2*n+1>=1 by NAT_1:11;
    then
A5: 2*n+1>0 by XXREAL_0:2;
    2*n+5>=5 by NAT_1:11;
    then
A6: 2*n+5>0 by XXREAL_0:2;
    Partial_Sums(s).(n+1)=1/12-1/(4*(2*n+1)*(2*n+3))+ s.(n+1) by A3,
SERIES_1:def 1
      .=1/12-1/(4*(2*n+1)*(2*n+3))+1/((2*(n+1)-1)*(2*(n+1)+1)*(2*(n+1)+3))
    by A1,A4
      .=1/12-1/(4*((2*n+1)*(2*n+3)))+1/((2*n+1)*(2*n+3)*(2*n+5))
      .=1/12-(1/4)*(1/((2*n+1)*(2*n+3)))+1/((2*n+1)*(2*n+3)*(2*n+5)) by
XCMPLX_1:102
      .=1/12-(1/4)*(1/((2*n+1)*(2*n+3)))+(1/((2*n+1)*(2*n+3)))*(1/(2*n+5))
    by XCMPLX_1:102
      .=1/12-1/((2*n+1)*(2*n+3))*(1/4-1/(2*n+5))
      .=1/12-1/((2*n+1)*(2*n+3))*((1*(2*n+5)-1*4)/(4*(2*n+5))) by A6,
XCMPLX_1:130
      .=1/12-(1/(2*n+1))*(1/(2*n+3))*((2*n+1)/(4*(2*n+5))) by XCMPLX_1:102
      .=1/12-(1/(2*n+1))*(1/(2*n+3))*(2*n+1)/(4*(2*n+5)) by XCMPLX_1:74
      .=1/12-(1/(2*n+3))/(4*(2*n+5)) by A5,XCMPLX_1:109
      .=1/12-1/((2*n+3)*(4*(2*n+5))) by XCMPLX_1:78
      .=1/12-1/(4*(2*(n+1)+1)*(2*(n+1)+3));
    hence thesis;
  end;
  Partial_Sums(s).(1+0)=Partial_Sums(s).0+s.(1+0) by SERIES_1:def 1
    .= s.0 + s.1 by SERIES_1:def 1
    .=0+s.1 by A1
    .=1/((2*1-1)*(2*1+1)*(2*1+3)) by A1
    .= 1/12-1/(4*(2*1+1)*(2*1+3));
  then
A7: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A7,A2);
  hence thesis;
end;
