Definition:Partial Function Equality

Let $$g: \N^k \to \N$$ and $$h: \N^k \to \N$$ be partial functions.

We write:
 * $$g \left({n_1, n_2, \ldots, n_k}\right) \approx h \left({n_1, n_2, \ldots, n_k}\right)$$

iff either:
 * both $$g \left({n_1, n_2, \ldots, n_k}\right)$$ and $$h \left({n_1, n_2, \ldots, n_k}\right)$$ are defined and equal, or:
 * neither $$g \left({n_1, n_2, \ldots, n_k}\right)$$ nor $$h \left({n_1, n_2, \ldots, n_k}\right)$$ are defined.

That is, iff $$g \left({n_1, n_2, \ldots, n_k}\right) = h \left({n_1, n_2, \ldots, n_k}\right)$$ wherever either are defined.

Thus, $$g$$ is equal to $$h$$, and we can write $$g = h$$, iff:
 * $$\forall x \in \N^k: g \left({x}\right) \approx h \left({x}\right)$$.