reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem Th48:
  (for n holds s.n=1/((n+1)^2)) implies for n holds (Partial_Sums
  s).n<=2-1/(n+1)
proof
  defpred X[Nat] means (Partial_Sums s).$1<=2-1/($1+1);
  assume
A1: for n holds s.n=1/((n+1)^2);
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume (Partial_Sums s).n<=2-1/(n+1);
    then
A3: (Partial_Sums s).n+1/((n+2)^2)<=2-1/(n+1)+1/((n+2)^2) by XREAL_1:7;
    n^2+3*n+3>n^2+3*n+2 by XREAL_1:8;
    then (n^2+3*n+3)/((n+1)*((n+2)^2))>((n+2)*(n+1))/(((n+2)^2)*(n+1)) by
XREAL_1:74;
    then (n^2+3*n+3)/((n+1)*((n+2)^2))>(n+2)/((n+2)*(n+2))by XCMPLX_1:91;
    then (n^2+3*n+3)/((n+1)*((n+2)^2))>((n+2)/(n+2))/(n+2) by XCMPLX_1:78;
    then (n^2+3*n+3)/((n+1)*((n+2)^2))>1/(n+2) by XCMPLX_1:60;
    then (-1)*((n^2+3*n+3)/((n+1)*((n+2)^2)))<(-1)*(1/(n+2)) by XREAL_1:69;
    then
A4: -(n^2+3*n+3)/((n+1)*((n+2)^2))+2<-1/(n+2)+2 by XREAL_1:8;
    (Partial_Sums s).(n+1) =(Partial_Sums s).n+s.(n+1) by SERIES_1:def 1
      .=(Partial_Sums s).n+1/((n+1+1)^2) by A1;
    then (Partial_Sums s).(n+1)<=2-(1/(n+1)-1/((n+2)^2)) by A3;
    then (Partial_Sums s).(n+1)<=2-(1*((n+2)^2)-1*(n+1))/((n+1)*((n+2)^2)) by
XCMPLX_1:130;
    hence thesis by A4,XXREAL_0:2;
  end;
  (Partial_Sums s).0=s.0 by SERIES_1:def 1
    .=1/((0+1)^2) by A1
    .=1;
  then
A5: X[0];
  for n holds X[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
