Function Obtained by Minimization from URM Computable Relations

Theorem
Let $\mathcal R$ be a URM computable $k+1$-ary relation on $\N^{k+1}$.

Let the function $f: \N^{k+1} \to \N$ be a URM computable function.

Let $g: \N^k \to \N$ be the function obtained by minimization from $f$ thus:
 * $g \left({n_1, n_2, \ldots, n_k}\right) \approx \mu y \mathcal R \left({n_1, n_2, \ldots, n_k, y}\right)$

Then $g$ is also URM computable.

Proof
From Minimization on Relation Equivalent to Minimization on Function, minimization on $\mathcal R$ is equivalent to minimization on $\overline{\operatorname{sgn}} \circ \chi_\mathcal R$.

We have that a Primitive Recursive Function is URM Computable.

By definition, if $\mathcal R$ is URM computable then so is its characteristic function $\chi_\mathcal R$.

We have that $\overline{\operatorname{sgn}}$ is primitive recursive and thus URM computable.

Thus, from Function Obtained by Substitution from URM Computable Functions, $\overline{\operatorname{sgn}} \circ \chi_\mathcal R$ is URM computable.