Currently, the proof of Theorem 25.9 states that

$$h(f^M(\text{Val}^M_s(t_1),\dots,\text{Val}^M_s(t_n)))$$

is equal to

$$h(f^M(\text{Val}^{M'}_{h \circ s}(t_1),\dots,\text{Val}^{M'}_{h \circ s}(t_n))$$

by the induction hypothesis that

$$h(\text{Val}^M_s(t_i))=\text{Val}^{M'}_{h \circ s}(t_i).$$

However, I'm having difficulty seeing how that is the case.

If instead we use item (5) of Definition 25.8, then we get

$$f^{M'}(h(\text{Val}^M_s(t_1)),\dots,h(\text{Val}^M_s(t_n)))$$

and by the induction hypothesis,

$$f^{M'}(\text{Val}^{M'}_{h \circ s}(t_1),\dots,\text{Val}^{M'}_{h \circ s}(t_n)).$$