The question is concerning the solution to exercise 3.5. In the exercise, they ask us to prove that any finite set of simultaneous equations $x_iy_1...y_n=Z_i$, where $1\leq i\leq k$, can be solved for each $x_i$ in both $\lambda$ and CL.
The situation with just one equation I understand as it was presented in Corollary 3.3.1. There they say that a solution $X$ is a term such that $Xy_1...y_n=_{\beta,w}[X/x]Z$. And they give $X\equiv Y(\lambda xy_1...y_n.Z)$ as a solution. (where $Y$ is a fixed point combinator.) This makes sense to me.
However, in the solution given to the exercise in the book, they first define $X$ to be the solution to $xy_1...y_n=D^{(k)}Z_1...Z_k$, where $D^{(k)}$ is a k-tuple combinator, and then they say that $X_i\equiv\lambda y_1...y_k.D^{(k)}_i(Xy_1...y_k)$ are the solutions to the set of equations, where $D^{(k)}_i$ is the i-th projection of $D^{(k)}$.
First of all I suspect the subscript k's in $X_i$ should be n's, but even then this confuses me, for if I've understood correctly, what we get when we plug this into the equations over is that $X_iy_1...y_n=_{\beta,w}[X/x]Z_i$. To me this doesn't look like a solution to simultaneous equations. I would have thought that it would look more like $$X_iy_1...y_n=_{\beta,w}[X_1/x_1,...X_i/x_i,...,X_k/x_k]Z_i$$ for $1\leq i\leq k$.
One obvious problem to me is that the variable $x$ in the solution given in the book doesn't seem to have any relation to the $x_i$'s. Other than that I could accept the way they did it as a definition different from what I would intuitively think of. As long as I understood what the relation between $x$ and $x_i$ was, I'd be totally fine with this. But right after the exercise they prove the "Double fixed-point theorem" where they seem to use the definition of simultaneous solutions I proposed! This bugs me a lot, so if anyone has any insight to share on this it would be extremely much appreciated!