Addition in Minimally Inductive Set is Unique

Theorem
Let $\omega$ be the minimally inductive set.

Let $A: \omega \times \omega \to \omega$ be the mapping defined as the addition operation:


 * $\forall \tuple {x, y} \in \omega \times \omega: \map A {x, y} = \begin {cases} x & : y = 0 \\ \paren {\map A {x, r} }^+ & : y = r^+ \end {cases}$

where $r^+$ is the successor set of $r$.

Then $A$ exists and is unique.

Proof
Lt $c \in \omega$ be arbitrary.

Let $g: \omega \times \omega \to \omega$ be the mapping defined as: