We will construct a partition $\lambda$ that fails to satisfy Equation .

Take a prime $p$, larger than any hook length in $\mu$. Consider now the partition $\lambda$ whose core is the empty partition and whose quotient consists of $p$ copies of $\mu$ (many other choices are possible). Then the valuation of the LHS of at $p$ is negative (and at least -$p$), our contradiction.

**$\Rightarrow$ :** Pick a prime $p$. Our goal is to prove that the valuation of at $p$ is positive. Because is a quotient, this valuation is given as a linear combination of the counts of hooks of $\lambda$ divisible by the $\gamma_k$ and $\delta_l$. This is thus a linear combination of the valuations at $p$ of the hook product of $\nu$, for $\nu$ running through the various $\gamma_k$ and $\delta_l$ quotients of $\lambda$. Each such valuation can be expressed, thanks to Lemma \[lemma.tower\], as a(n effectively finite) sum over the labels in the $p$-core tower of $\nu$. We now swap this linear combination over $\nu$ and the infinite sum over label positions in a $p$-core tower. We then get a sum over positions in the $p$-core tower of an expression of the form ($\mu$ runs 