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

theorem
  ReciProducts 0 = { 1 }
  proof
    the set of all 1 / Product Sgm X where X is Subset of SetPrimes 0 = {1}
    proof
T1:   the set of all 1 / Product Sgm X where X is Subset of SetPrimes 0 c= {1}
      proof
        let x be object;
        assume x in the set of all
          1 / Product Sgm X where X is Subset of SetPrimes 0; then
        consider X being Subset of SetPrimes 0 such that
C1:     x = 1 / Product Sgm X;
        Sgm X = {} by FINSEQ_3:43;
        hence thesis by TARSKI:def 1,C1,RVSUM_1:94;
      end;
      {1} c=
        the set of all 1 / Product Sgm X where X is Subset of SetPrimes 0
      proof
        let x be object;
        assume x in {1}; then
S1:     x = 1 / Product Sgm {} by FINSEQ_3:43,RVSUM_1:94,TARSKI:def 1;
        {} c= SetPrimes 0;
        hence thesis by S1;
      end;
      hence thesis by T1,TARSKI:2;
    end;
    hence thesis;
  end;
