 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem ::: Similar bound in BASEL series, counted independently
  (Partial_Sums Basel-seq).n < 5 / 3
  proof
    per cases by NAT_2:28;
    suppose n is non trivial; then
      Sum (Basel-seq, n) < 5 / 3 by Important;
      hence thesis;
    end;
    suppose n = 0; then
      (Partial_Sums Basel-seq).n = Basel-seq.0 by SERIES_1:def 1
        .= 1 / (0 ^2) by BASEL_1:31
        .= 0;
      hence thesis;
    end;
    suppose
F:    n = 1;
      (Partial_Sums Basel-seq).(0+1) = (Partial_Sums Basel-seq).0 + Basel-seq.1
        by SERIES_1:def 1
      .= Basel-seq.0 + Basel-seq.1 by SERIES_1:def 1
      .= 1 / (0 ^2) + Basel-seq.1 by BASEL_1:31
      .= 0 + Basel-seq.1
      .= 0 + 1 / (1 ^2) by BASEL_1:31
      .= 1;
      hence thesis by F;
    end;
  end;
