Prove that the coding $\pi_k$ is effective
Since $\pi_k$ is bijective, for every $x∈N$, there exists a unique tuple $(x_1, ..., x_k)∈N^k$ such that $\pi_k(x_1,...,x_n) = x$. So, how can we define $J^k_i$ ? A solution would be to use a coding that is already known to be effective, let $\alpha_k$ be that effective coding and $\beta^k_1,..., \beta^k_k$ the associated decoding functions. Let $x∈N$, we can put : $$J^k_i(x) = \beta_i^k(\mu y(\ \pi_k(\ \beta^k_1(y)\ , ...,\ \beta^k_k(y)) = x))$$

For $\alpha_k$, we can take the cantor tuple function, which is known to be primitive recursive.

Of course, that's not a proper definition, we need mathematical induction to throw away those horrible dots.