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

theorem
  (for n holds s.n = (n|^2*4|^n)/((n+1)*(n+2))) implies for n st n>=1
  holds Partial_Sums(s).n =2/3+((n-1)*4|^(n+1))/(3*(n+2))
proof
  defpred X[Nat] means Partial_Sums(s).$1=2/3+(($1-1)*4|^($1+1))/(3*($1+2));
  assume
A1: for n holds s.n = (n|^2*4|^n)/((n+1)*(n+2));
  then
A2: s.0 = (0|^2*4|^0)/((0+1)*(0+2)) .= (0*4|^0)/(1*2) by NEWTON:11
    .= 0;
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 =2/3+((n-1)*4|^(n+1))/(3*(n+2));
    n+2>=2 by NAT_1:11;
    then
A5: n+2>0 by XXREAL_0:2;
    n+3>=3 by NAT_1:11;
    then
A6: n+3>0 by XXREAL_0:2;
    Partial_Sums(s).(n+1) =2/3+((n-1)*4|^(n+1))/(3*(n+2))+ s.(n+1) by A4,
SERIES_1:def 1
      .=2/3+((n-1)*4|^(n+1))/(3*(n+2)) +((n+1)|^2*4|^(n+1))/(((n+1)+1)*((n+1
    )+2)) by A1
      .=2/3+((n-1)*4|^(n+1)*(n+3))/(3*(n+2)*(n+3)) +((n+1)|^2*4|^(n+1))/((n+
    2)*(n+3)) by A6,XCMPLX_1:91
      .=2/3+((n-1)*4|^(n+1)*(n+3))/(3*(n+2)*(n+3)) +((n+1)|^2*4|^(n+1)*3)/((
    n+2)*(n+3)*3) by XCMPLX_1:91
      .=2/3+(((n-1)*4|^(n+1)*(n+3))/(3*(n+2)*(n+3)) +((n+1)|^2*4|^(n+1)*3)/(
    3*(n+2)*(n+3)))
      .=2/3+(((n-1)*4|^(n+1)*(n+3))+((n+1)|^2*4|^(n+1)*3)) /(3*(n+2)*(n+3))
    by XCMPLX_1:62
      .=2/3+(((n-1)*(n+3)+(n+1)|^2*3)*4|^(n+1))/(3*(n+2)*(n+3))
      .=2/3+(((n-1)*(n+3)+(n+1)*(n+1)*3)*4|^(n+1))/(3*(n+2)*(n+3)) by
WSIERP_1:1
      .=2/3+(4*4|^(n+1)*n*(n+2))/(3*(n+3)*(n+2))
      .=2/3+(4|^(n+1)*4*n)/(3*(n+3)) by A5,XCMPLX_1:91
      .=2/3+((n+1-1)*4|^(n+1+1))/(3*(n+1+2)) by NEWTON:6;
    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*4|^1)/((1+1)*(1+2)) by A1,A2
    .=(1*4|^1)/((1+1)*(1+2))
    .=(1*4)/((1+1)*(1+2))
    .=2/3+((1-1)*4|^(1+1))/(3*(1+2));
  then
A7: X[1];
  for n be Nat st n>=1 holds X[n] from NAT_1:sch 8(A7,A3);
  hence thesis;
end;
