Solution
= Solution
The first two clauses now require $x=2z$ and $y=2z$. Thus they already imply $x=y$. Choose, for example, $x=1,y=2$; no <natural number> $z$ can satisfy both equations. Consequently the <existential quantifier> fails for this pair, irrespective of the universally quantified final clause, and
$$
\boxed{\mathbf N''\not\models\psi.}
$$
In fact even $x=y=1$ fails, since $1=2z$ has no solution among the <natural numbers>.