reserve a, b, n for Nat,
  r for Real,
  f for FinSequence of REAL;
reserve p for Prime;

theorem
  for f being FinSequence of SetPrimes st p divides Product f holds p in rng f
proof
  let f be FinSequence of SetPrimes;
  defpred P[FinSequence] means
  ex f being FinSequence of SetPrimes st f = $1 &
  for p being Prime st p divides Product f holds p in rng f;
A1: now
    let f be FinSequence of SetPrimes, n be Element of SetPrimes;
    set f1 = f^<*n*>;
    assume
A2: P[f];
    thus P[f1]
    proof
      reconsider nn = n as Nat;
      reconsider ff = f as FinSequence of NAT;
      reconsider f2 = f1 as FinSequence of SetPrimes;
      take f2;
      thus f2 = f1;
      let p be Prime;
      assume p divides Product f2;
      then
A3:   p divides Product ff * n by RVSUM_1:96;
      per cases by A3,NEWTON:80;
      suppose
A4:     p divides Product f;
A5:     rng f c= rng f1 by FINSEQ_1:29;
        p in rng f by A2,A4;
        hence thesis by A5;
      end;
      suppose
A6:     p divides nn;
        nn is prime by NEWTON:def 6;
        then p = 1 or p = n by A6;
        then p in {n} by INT_2:def 4,TARSKI:def 1;
        then
A7:     p in rng <*n*> by FINSEQ_1:38;
        rng <*n*> c= rng f1 by FINSEQ_1:30;
        hence thesis by A7;
      end;
    end;
  end;
A8: P[<*>SetPrimes]
  proof
    take <*>SetPrimes;
    thus thesis by INT_2:def 4,NAT_D:7,RVSUM_1:94;
   end;
  for p being FinSequence of SetPrimes holds P[p] from FINSEQ_2:sch 2(A8,A1);
  then P[f];
  hence thesis;
end;
