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

theorem Th14:
  Product (f|^(b+1)) = Product (f|^b) * Product f
proof
  defpred P[FinSequence of REAL] means for b being Nat holds
  Product ($1|^(b+1)) = Product ($1|^b) * Product $1;
A1: now
    let p be FinSequence of REAL, x being Element of REAL such that
A2: P[p];
    thus P[p^<*x*>]
    proof
      set p1 = p^<*x*>;
      let b be Nat;
      reconsider xb1 = x|^(b+1), xb = x|^b as Real;
      p1 |^ (b+1) = (p|^(b+1))^(<*x*>|^(b+1)) by Th13;
      hence Product (p1|^(b+1)) = Product (p|^(b+1)) * Product (<*x*>|^(b+1))
      by RVSUM_1:97
        .= (Product (p|^b) * Product p) * Product (<*x*>|^(b+1)) by A2
        .= (Product (p|^b) * Product p) * Product (<*xb1*>) by Th12
        .= (Product (p|^b) * Product p) * x|^(b+1)
        .= (Product (p|^b) * Product p) * (x|^b * x) by NEWTON:6
        .= Product (p|^b) * x|^b * x * Product p
        .= Product (p|^b ^ <*xb*>) * x * Product p by RVSUM_1:96
        .= Product (p|^b ^ (<*x*>|^b)) * x * Product p by Th12
        .= Product (p1|^b) * x * Product p by Th13
        .= Product (p1|^b) * (Product p * x)
        .= Product (p1|^b) * Product p1 by RVSUM_1:96;
    end;
  end;
A3: P[<*>REAL]
  proof
    set f = <*>REAL;
    let b be Nat;
    thus Product (f|^(b+1)) = 1 by Th11,RVSUM_1:94
      .= Product (f|^b) * Product f by Th11,RVSUM_1:94;
  end;
  for p being FinSequence of REAL holds P[p] from FINSEQ_2:sch 2(A3, A1);
  hence thesis;
end;
