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

theorem Important:
  for n being non trivial Nat holds
    Sum (Basel-seq, n) < 5 / 3
  proof
    let n be non trivial Nat;
Z2: Sum (Basel-seq, n)
         = (Partial_Sums Basel-seq).(0 + 1) + ((Partial_Sums Basel-seq).n -
          (Partial_Sums Basel-seq).1)
        .= ((Partial_Sums Basel-seq).0 + Basel-seq.1) +
          ((Partial_Sums Basel-seq).n -
          (Partial_Sums Basel-seq).1) by SERIES_1:def 1
        .= Basel-seq.0 + Basel-seq.1 + ((Partial_Sums Basel-seq).n -
          (Partial_Sums Basel-seq).1) by SERIES_1:def 1
        .= 1 / (0 ^2) + Basel-seq.1 + Sum (Basel-seq, n, 1) by BASEL_1:31
        .= 1 / 0 + 1 / (1 ^2) + Sum (Basel-seq, n, 1) by BASEL_1:31
        .= 1 + Sum (Basel-seq, n, 1);
Z5: Sum (Basel-seq, n) < 1 + Sum (Reci-seq1, n, 1) by Z2,XREAL_1:8,Impor2;
    1 + Sum (Reci-seq1, n, 1) < 1 + 2 / 3 by XREAL_1:8,Seq3;
    hence thesis by Z5,XXREAL_0:2;
  end;
