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

theorem
  (for n st n>=1 holds s.n = 1/((3*n-2)*(3*n+1)*(3*n+4)) & s.0=0)
  implies for n st n>=1 holds Partial_Sums(s).n =1/24-1/(6*(3*n+1)*(3*n+4))
proof
  defpred X[Nat] means Partial_Sums(s).$1= 1/24-1/(6*(3*$1+1)*(3*$1+4));
  assume
A1: for n st n>=1 holds s.n =1/((3*n-2)*(3*n+1)*(3*n+4)) & 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 =1/24-1/(6*(3*n+1)*(3*n+4));
A4: n+1>=1 by NAT_1:11;
    3*n+1>=1 by NAT_1:11;
    then
A5: 3*n+1>0 by XXREAL_0:2;
    3*n+7>=7 by NAT_1:11;
    then
A6: 3*n+7>0 by XXREAL_0:2;
    Partial_Sums(s).(n+1)=1/24-1/(6*(3*n+1)*(3*n+4))+ s.(n+1) by A3,
SERIES_1:def 1
      .=1/24-1/(6*(3*n+1)*(3*n+4))+1/((3*(n+1)-2)*(3*(n+1)+1)*(3*(n+1)+4))
    by A1,A4
      .=1/24-1/(6*((3*n+1)*(3*n+4)))+1/(((3*n+1)*(3*n+4))*(3*n+7))
      .=1/24-(1/6)*(1/((3*n+1)*(3*n+4)))+1/(((3*n+1)*(3*n+4))*(3*n+7)) by
XCMPLX_1:102
      .=1/24-(1/6)*(1/((3*n+1)*(3*n+4)))+(1/((3*n+1)*(3*n+4)))*(1/(3*n+7))
    by XCMPLX_1:102
      .=1/24-(1/((3*n+1)*(3*n+4)))*(1/6-1/(3*n+7))
      .=1/24-(1/((3*n+1)*(3*n+4)))*((1*(3*n+7)-1*6)/(6*(3*n+7))) by A6,
XCMPLX_1:130
      .=1/24-(1/(3*n+4))*(1/(3*n+1))*((3*n+1)/(6*(3*n+7))) by XCMPLX_1:102
      .=1/24-(1/(3*n+4))*(1/(3*n+1))*(3*n+1)/(6*(3*n+7)) by XCMPLX_1:74
      .=1/24-(1/(3*n+4))/(6*(3*n+7)) by A5,XCMPLX_1:109
      .=1/24-1/((3*n+4)*(6*(3*n+7))) by XCMPLX_1:78
      .=1/24-1/(6*(3*(n+1)+1)*(3*(n+1)+4));
    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/((3*1-2)*(3*1+1)*(3*1+4)) by A1
    .=1/24-1/(6*(3*1+1)*(3*1+4));
  then
A7: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A7,A2);
  hence thesis;
end;
