Single Instruction URM Programs/Projection Function

Theorem
The projection functions $\operatorname{pr}^k_j: \N^k \to \N$, defined as:
 * $\forall j \in \left[{1 \,.\,.\, k}\right]: \forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: \operatorname{pr}^k_j \left({\left({n_1, n_2, \ldots, n_k}\right)}\right) = n_j$

are each URM computable by a single-instruction URM program.

Proof
The projection functions are computed by the following URM program:

The input $\left({n_1, n_2, \ldots, n_j, \ldots, n_k}\right)$ is in $R_1, R_2, \ldots, R_j, \ldots, R_k$ when the program starts.

The program copies $r_j$ to $r_1$ and then stops.

The output $n_j$ is in $R_1$ when the program terminates.