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

theorem
  (for n holds s.n = (n+2)/(n*(n+1)*(n+3))) implies for n st n>=1 holds
  Partial_Sums(s).n = 29/36-1/(n+3)-3/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+2)*(n+3))
proof
  defpred X[Nat] means Partial_Sums(s).$1 = 29/36-1/($1+3)-3/(2*($1+2)*($1+3))
  -4/(3*($1+1)*($1+2)*($1+3));
  assume
A1: for n holds s.n = (n+2)/(n*(n+1)*(n+3));
  then
A2: s.0 = (0+2)/(0*(0+1)*(0+3)) .= 0 by XCMPLX_1:49;
A3: for n be Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
    n>=1 and
A4: Partial_Sums(s).n = 29/36-1/(n+3)-3/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+
    2) *(n+3));
    n+4>=4 by NAT_1:11;
    then
A5: n+4>0 by XXREAL_0:2;
    n+1>=1+0 by NAT_1:11;
    then
A6: n+1>0 by NAT_1:13;
    then
A7: 2*(n+1)>0 by XREAL_1:129;
    n+2>=2 by NAT_1:11;
    then
A8: n+2>0 by XXREAL_0:2;
    then
A9: (n+1)*(n+2)>0 by A6,XREAL_1:129;
    then
A10: (n+1)*(n+2)*3>0 by XREAL_1:129;
    n+3>=3 by NAT_1:11;
    then
A11: n+3>0 by XXREAL_0:2;
    then (n+1)*(n+2)*(n+3)>0 by A9,XREAL_1:129;
    then
A12: (n+1)*(n+2)*(n+3)*6>0 by XREAL_1:129;
    Partial_Sums(s).(n+1) =29/36-1/(n+3)-3/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+
    2)*(n+3))+s.(n+1) by A4,SERIES_1:def 1
      .=29/36-1/(n+3)-3/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+2)*(n+3))+(n+1+2)/((n+
    1) *(n+1+1)*(n+1+3)) by A1
      .=29/36-(1*(n+2))/((n+3)*(n+2))-3/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+2) *(n
    +3))+(n+3)/((n+1)*(n+2)*(n+4)) by A8,XCMPLX_1:91
      .=29/36-((n+2)*2)/((n+2)*(n+3)*2)-3/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+2) *
    (n+3))+(n+3)/((n+1)*(n+2)*(n+4)) by XCMPLX_1:91
      .=29/36-(((n+2)*2)/((n+2)*(n+3)*2)+3/(2*(n+2)*(n+3)))-4/(3*(n+1)*(n+2)
    *(n+3))+(n+3)/((n+1)*(n+2)*(n+4))
      .=29/36-(((n+2)*2)+3)/(2*(n+2)*(n+3))-4/(3*(n+1)*(n+2) *(n+3))+(n+3)/(
    (n+1)*(n+2)*(n+4)) by XCMPLX_1:62
      .=29/36-((n*2+7)*(n+1))/(2*(n+2)*(n+3)*(n+1))-4/(3*(n+1)*(n+2)*(n+3))
    +(n+3)/((n+1)*(n+2)*(n+4)) by A6,XCMPLX_1:91
      .=29/36-((n*2+7)*(n+1)*3)/(2*(n+2)*(n+3)*(n+1)*3)-4/(3*(n+1)*(n+2)*(n+
    3)) +(n+3)/((n+1)*(n+2)*(n+4)) by XCMPLX_1:91
      .=29/36-((n*2+7)*(n+1)*3)/(2*(n+2)*(n+3)*(n+1)*3)-(4*2)/(3*(n+1)*(n+2)
    *(n+3) *2)+(n+3)/((n+1)*(n+2)*(n+4)) by XCMPLX_1:91
      .=29/36-(((n*2+7)*(n+1)*3)/(6*(n+2)*(n+3)*(n+1))+8/(6*(n+1)*(n+2)*(n+3
    ))) +(n+3)/((n+1)*(n+2)*(n+4))
      .=29/36-(((n*2+7)*(n+1)*3*(n+4))/(6*(n+2)*(n+3)*(n+1)*(n+4))+ 8/(6*(n+
    1)*(n+2)*(n+3)))+(n+3)/((n+1)*(n+2)*(n+4)) by A5,XCMPLX_1:91
      .=29/36-(((n*2+7)*(n+1)*3*(n+4))/(6*(n+1)*(n+2)*(n+3)*(n+4))+(8*(n+4))
    /(6 *(n+1)*(n+2)*(n+3)*(n+4)))+(n+3)/((n+1)*(n+2)*(n+4))by A5,XCMPLX_1:91
      .=29/36-((n*2+7)*(n+1)*3*(n+4)+8*(n+4))/(6*(n+1)*(n+2)*(n+3)*(n+4))+(n
    +3) /((n+1)*(n+2)*(n+4)) by XCMPLX_1:62
      .=29/36-((n*2+7)*(n+1)*3*(n+4)+8*(n+4))/(6*(n+1)*(n+2)*(n+3)*(n+4))+((
    n+3) *6)/((n+1)*(n+2)*(n+4)*6) by XCMPLX_1:91
      .=29/36-((n*2+7)*(n+1)*3*(n+4)+8*(n+4))/(6*(n+1)*(n+2)*(n+3)*(n+4))+((
    n+3) *6*(n+3))/((n+1)*(n+2)*(n+4)*6*(n+3)) by A11,XCMPLX_1:91
      .=29/36-(((n*2+7)*(n+1)*3*(n+4)+8*(n+4))/(6*(n+1)*(n+2)*(n+3)*(n+4)) -
    ((n+3)*6*(n+3))/((n+1)*(n+2)*(n+4)*6*(n+3)))
      .=29/36-(((n*2+7)*(n+1)*3*(n+4)+8*(n+4))-((n+3)*6*(n+3))) /(6*(n+1)*(n
    +2)*(n+3)*(n+4)) by XCMPLX_1:120
      .=29/36-(6*(n+1)*(n+2)*(n+3)+9*(n+1)*(n+2)+8*(n+1)) /(6*(n+1)*(n+2)*(n
    +3)*(n+4))
      .=29/36-((6*(n+1)*(n+2)*(n+3)+9*(n+1)*(n+2))/(6*(n+1)*(n+2)*(n+3)*(n+4
    )) +(8*(n+1))/(6*(n+1)*(n+2)*(n+3)*(n+4))) by XCMPLX_1:62
      .=29/36-((1*(6*(n+1)*(n+2)*(n+3)))/((n+4)*(6*(n+1)*(n+2)*(n+3))) +(9*(
n+1)*(n+2))/(6*(n+1)*(n+2)*(n+3)*(n+4)) +(8*(n+1))/(6*(n+1)*(n+2)*(n+3)*(n+4)))
    by XCMPLX_1:62
      .=29/36-(1/(n+4)+(3*(3*(n+1)*(n+2)))/(2*(n+3)*(n+4)*(3*(n+1)*(n+2))) +
    (8*(n+1))/(6*(n+1)*(n+2)*(n+3)*(n+4))) by A12,XCMPLX_1:91
      .=29/36-(1/(n+4)+3/(2*(n+3)*(n+4)) +(4*(2*(n+1)))/(3*(n+2)*(n+3)*(n+4)
    *(2*(n+1)))) by A10,XCMPLX_1:91
      .=29/36-(1/(n+4)+3/(2*(n+3)*(n+4))+4/(3*(n+2)*(n+3)*(n+4))) by A7,
XCMPLX_1:91
      .=29/36-1/(n+1+3)-3/(2*(n+1+2)*(n+1+3))-4/(3*(n+1+1)*(n+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
    .=(1+2)/(1*(1+1)*(1+3)) by A1,A2
    .=29/36-1/(1+3)-3/(2*(1+2)*(1+3))-4/(3*(1+1)*(1+2)*(1+3));
  then
A13: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A13,A3);
  hence thesis;
end;
