We have $[W_a]_r = \text{pr}_{X,*}(W \cdot X \times a)$ and similarly for $[W_b]_r$, see Lemma \ref{lemma-rational-equivalence-and-intersection}. Thus we reduce to showing $$ V \cdot \text{pr}_{X,*}( W \cdot X \times a) \sim_{rat} V \cdot \text{pr}_{X,*}( W \cdot X\times b). $$ Applying the projection formula Lemma \ref{lemma-projection-formula-flat} we get $$ V \cdot \text{pr}_{X,*}( W \cdot X\times a) = \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) $$ and similarly for $b$. Thus we reduce to showing $$ \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) \sim_{rat} \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times b)) $$ If $V \times \mathbf{P}^1$ intersects $W$ properly, then associativity for the intersection multiplicities (Lemma \ref{lemma-associative}) gives $V \times \mathbf{P}^1 \cdot (W \cdot X\times a) = (V \times \mathbf{P}^1 \cdot W) \cdot X \times a$ and similarly for $b$.  Thus we reduce to showing $$ \text{pr}_{X,*}((V \times \mathbf{P}^1 \cdot W) \cdot X\times a) \sim_{rat} \text{pr}_{X,*}((V \times \mathbf{P}^1 \cdot W) \cdot X\times b) $$ which is true by Lemma \ref{lemma-rational-equivalence-and-intersection}. \medskip\noindent The argument above does not quite work. The obstruction is that we do not know that $V \times \mathbf{P}^1$ and $W$ intersect properly. We only know that $V$ and $W_a$ and $V$ and $W_b$ intersect properly. Let $Z_i$, $i \in I$ be the irreducible components of $V \times \mathbf{P}^1 \cap W$. Then we know that $\dim(Z_i) \geq r + 1 + s + 1 - n - 1 = r + s + 1 - n$ where $n = \dim(X)$, see Lemma \ref{lemma-intersect-in-smooth}. Since we have assumed that $V$ and $W_a$ intersect properly, we see that $\dim(Z_{i, a}) = r + s - n$ or $Z_{i, a} = \emptyset$. On the other hand, if $Z_{i, a} \not = \emptyset$, then $\dim(Z_{i, a}) \geq \dim(Z_i) - 1 = r + s - n$. It follows that $\dim(Z_i) = r + s + 1 - n$ if $Z_i$ meets $X \times a$ and in this case $Z_i \to \mathbf{P}^1$ is surjective. Thus we may write $I = I' \amalg I''$ where $I'$ is the set of $i \in I$ such that $Z_i \to \mathbf{P}^1$ is surjective and $I''$ is the set of $i \in I$ such that $Z_i$ lies over a closed point $t_i \in \mathbf{P}^1$ with $t_i \not = a$ and $t_i \not = b$. Consider the cycle $$ \gamma = \sum\nolimits_{i \in I'} e_i [Z_i] $$ where we take $$ e_i = \sum\nolimits_p (-1)^p \text{length}_{\mathcal{O}_{X \times \mathbf{P}^1, Z_i}} \text{Tor}_p^{\mathcal{O}_{X \times \mathbf{P}^1, Z_i}}( \mathcal{O}_{V \times \mathbf{P}^1, Z_i}, \mathcal{O}_{W, Z_i}) $$ We will show that $\gamma$ can be used as a replacement for the intersection product of $V \times \mathbf{P}^1$ and $W$. \medskip\noindent We will show this using associativity of intersection products in exactly the same way as above. Let $U = \mathbf{P}^1 \setminus \{t_i, i \in I''\}$. Note that $X \times a$ and $X \times b$ are contained in $X \times U$. The subvarieties $$ V \times U,\quad W_U,\quad X \times a\quad\text{of}\quad X \times U $$ intersect transversally pairwise by our choice of $U$ and moreover $\dim(V \times U \cap W_U \cap X \times a) = \dim(V \cap W_a)$ has the expected dimension. Thus we see that $$ V \times U \cdot (W_U \cdot X \times a) = (V \times U \cdot W_U) \cdot X \times a $$ as cycles on $X \times U$ by Lemma \ref{lemma-associative}. By construction $\gamma$ restricts to the cycle $V \times U \cdot W_U$ on $X \times U$. Trivially, $V \times \mathbf{P}^1 \cdot (W \times X \times a)$ restricts to $V \times U \cdot (W_U \cdot X \times a)$ on $X \times U$. Hence $$ V \times \mathbf{P}^1 \cdot (W \cdot X \times a) = \gamma \cdot X \times a $$ as cycles on $X \times \mathbf{P}^1$ (because both sides are contained in $X \times U$ and are equal after restricting to $X \times U$ by what was said before). Since we have the same for $b$ we conclude \begin{align*} V \cdot [W_a] & = \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times a)) \\ & = \text{pr}_{X, *}(\gamma \cdot X \times a) \\ & \sim_{rat}  \text{pr}_{X, *}(\gamma \cdot X \times b) \\ & = \text{pr}_{X,*}(V \times \mathbf{P}^1 \cdot (W \cdot X\times b)) \\ & = V \cdot [W_b] \end{align*} The first and the last equality by the first paragraph of the proof, the second and penultimate equalities were shown in this paragraph, and the middle equivalence is Lemma \ref{lemma-rational-equivalence-and-intersection}.