Natural Number Multiplication is Unique

Theorem
Let $\N$ denote the natural numbers.

Let $M: \N \times \N \to \N$ be the mapping defined as the multiplication operation:


 * $\forall \tuple {x, y} \in \N \times \N: \map M {x, y} = \begin {cases} 0 & : y = 0 \\ \paren {\map M {x, r} } + r & : y = r + 1 \end {cases}$

where $+$ denotes natural number addition.

Then $M$ exists and is unique.