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

theorem Crucial5X:
  for n being Nat holds
    ln.((Partial_Product Reci-seq2).n) <= (Partial_Sums ReciPrime).n
  proof
    let n be Nat;
    defpred P[Nat] means
    ln.((Partial_Product Reci-seq2).$1) <=
      ((Partial_Sums ReciPrime).$1);
B1: (Partial_Product Reci-seq2).0 = Reci-seq2.0 by SERIES_3:def 1
      .= 1 + 1 / 2 by MOEBIUS2:8,My3Def;
    (Partial_Sums ReciPrime).0 = ReciPrime.0 by SERIES_1:def 1
      .= 1 / 2 by MOEBIUS2:8,MOEBIUS2:def 1; then
A1: P[0] by B1,Diesel1;
A2: for k being Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume
B1:   P[k];
C1:   (Partial_Product Reci-seq2).k > 0 by GreaterProduct;
C2:   Reci-seq2.(k+1) > 0
      proof
        Reci-seq2.(k+1) = 1 + 1 / primenumber (k+1) by My3Def;
        hence thesis;
      end;
      (Partial_Product Reci-seq2).(k+1) =
        (Partial_Product Reci-seq2).k * (Reci-seq2.(k+1))
          by SERIES_3:def 1; then
      ln.((Partial_Product Reci-seq2).(k+1)) =
        ln.((Partial_Product Reci-seq2).k) + ln.((Reci-seq2.(k+1)))
          by LogAdd,C1,C2; then
D5:   ln.((Partial_Product Reci-seq2).(k+1)) <=
        ((Partial_Sums ReciPrime).k) + ln.((Reci-seq2.(k+1))) by XREAL_1:7,B1;
D3:   (Partial_Sums ReciPrime).(k+1) =
        (Partial_Sums ReciPrime).k + (ReciPrime.(k+1)) by SERIES_1:def 1;
      ln.((Reci-seq2.(k+1))) < ReciPrime.(k+1)
      proof
D1:     Reci-seq2.(k+1) = 1 + 1 / primenumber (k+1) by My3Def;
        ReciPrime.(k+1) = 1 / primenumber (k+1) by MOEBIUS2:def 1;
        hence thesis by D1,Diesel1;
      end; then
      ((Partial_Sums ReciPrime).k) + ln.((Reci-seq2.(k+1))) <
        ((Partial_Sums ReciPrime).k) + ReciPrime.(k+1) by XREAL_1:8;
      hence thesis by D5,XXREAL_0:2,D3;
    end;
    for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
