 reserve n,i,k,m for Nat;
 reserve p for Prime;
 reserve s, s1, s2 for Real_Sequence;

theorem
  for n be Nat st n+1 is Prime holds
    ReciProducts (n+1) = ReciProducts n \/
      {1 / Product Sgm X where X is Subset of SetPrimes (n+1) : n+1 in X}
  proof
    let n be Nat;
    assume
A0: n+1 is Prime;
T1: ReciProducts (n+1) c= ReciProducts n
    \/ {1 / Product Sgm X where X is Subset of SetPrimes (n+1) :
       n+1 in X}
    proof
      let x be object;
      assume x in ReciProducts (n+1); then
      consider X being Subset of SetPrimes (n+1) such that
A1:   x = 1 / Product Sgm X;
      X c= SetPrimes (n+1); then
A2:   X c= SetPrimes n \/ {n+1} by A0,PrimesSet2;
      per cases;
      suppose
        n+1 in X; then
        x in {1 / Product Sgm X where X is Subset of SetPrimes (n+1) :
          n+1 in X} by A1;
        hence thesis by XBOOLE_0:def 3;
      end;
      suppose
        not n+1 in X; then
        X c= SetPrimes n by A2,ZFMISC_1:135; then
        x in ReciProducts n by A1;
        hence thesis by XBOOLE_0:def 3;
      end;
    end;
    ReciProducts n \/
      {1 / Product Sgm X where X is Subset of SetPrimes (n+1) : n+1 in X}
        c= ReciProducts (n+1)
    proof
      let x be object;
      assume x in
      ReciProducts n
      \/ {1 / Product Sgm X where X is Subset of SetPrimes (n+1) :
       n+1 in X}; then
       per cases by XBOOLE_0:def 3;
       suppose Z1: x in ReciProducts n;
         ReciProducts n c= ReciProducts (n+1) by ReciSubset;
         hence thesis by Z1;
       end;
       suppose x in {1 / Product Sgm X where X is Subset of SetPrimes (n+1) :
         n+1 in X}; then
         consider X being Subset of SetPrimes (n+1) such that
B1:      x = 1 / Product Sgm X & n+1 in X;
         thus thesis by B1;
       end;
    end;
    hence thesis by T1,TARSKI:2;
  end;
