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

theorem
  for n being Nat holds
    ln.((Partial_Product Reci-seq2).(indexp n)) <= (Partial_Sums ReciPrime).n
  proof
    let n be Nat;
A1: ln.((Partial_Product Reci-seq2).(indexp n)) <=
      (Partial_Sums ReciPrime).(indexp n) by Crucial5X;
    for i being Nat holds ReciPrime.i >= 0
    proof
      let i be Nat;
      ReciPrime.i = 1 / primenumber i by MOEBIUS2:def 1;
      hence thesis;
    end; then
    (Partial_Sums ReciPrime).(indexp n) <=
      (Partial_Sums ReciPrime).n by Krzys1,CATALAN2:52;
    hence thesis by A1,XXREAL_0:2;
  end;
