 Suppose that \(f(\vec{x},z)\) is primitive recursive. Then the bounded sum \(g(\vec{x},y) = \Sigma_{i=0}^y f(\vec{x},i)\) and the bounded product \(h(\vec{x},y) = \Pi_{i=0}^y f(\vec{x},i)\) are both primitive recursive as they may be respectively defined as follows: 