The first identity says that $$\{h(x) : x \in \lambda\} \cup \{\mu_i - \mu_j : 1 \leq i < j \leq k\} = \bigcup_{i=1}^k \{j : 1 \leq j \leq \mu_i\}$$ as multisets. The second identity then follows.

The first identity says that $$\{h(x) : x \in \lambda\} \cup \{\mu_i - \mu_j : 1 \leq i < j \leq k\} = \bigcup_{i=1}^k \{j : 1 \leq j \leq \mu_i\}$$ as multisets. The second identity then follows because the multisets of exponents are the same on the LHS and RHS.