Pick a closed immersion $X \subset \mathbf{P}^N$. By linearity it suffices to prove the result when $\beta = [Z]$ for some $s$-dimensional closed subvariety $Z \subset X$ which intersects $\alpha$ properly. The condition $\alpha \sim_{rat} 0$ means there are finitely many $(r + 1)$-dimensional closed subvarieties $W_i \subset X \times \mathbf{P}^1$ such that $$ \alpha = \sum [W_{i, a_i}]_r - [W_{i, b_i}]_r $$ for some pairs of points $a_i, b_i$ of $\mathbf{P}^1$. Let $W_{i, a_i}^t$ and $W_{i, b_i}^t$ be the irreducible components of $W_{i, a_i}$ and $W_{i, b_i}$. We will use induction on the maximum $d$ of the integers $$ \dim(Z \cap W_{i, a_i}^t),\quad \dim(Z \cap W_{i, b_i}^t) $$ The main problem in the rest of the proof is that although we know that $Z$ intersects $\alpha$ properly, it may not be the case that $Z$ intersects the ``intermediate'' varieties $W_{i, a_i}^t$ and $W_{i, b_i}^t$ properly, i.e., it may happen that $d >  r + s - \dim(X)$. \medskip\noindent Base case: $d = r + s - \dim(X)$. In this case all the intersections of $Z$ with the $W_{i, a_i}^t$ and $W_{i, b_i}^t$ are proper and the desired result follows from Lemma \ref{lemma-well-defined-special-case}, because it applies to show that $[Z] \cdot [W_{i, a_i}]_r \sim_{rat} [Z] \cdot [W_{i, b_i}]_r$ for each $i$. \medskip\noindent Induction step: $d >  r + s - \dim(X)$. Apply Lemma \ref{lemma-moving} to $Z \subset X$ and the family of subvarieties $\{W_{i, a_i}^t, W_{i, b_i}^t\}$. Then we find a closed subvariety $C \subset \mathbf{P}^N$ intersecting $X$ properly such that $$ C \cdot X = [Z] + \sum m_j [Z_j] $$ and such that $$ \dim(Z_j \cap W_{i, a_i}^t) \leq \dim(Z \cap W_{i, a_i}^t),\quad \dim(Z_j \cap W_{i, b_i}^t) \leq \dim(Z \cap W_{i, b_i}^t) $$ with strict inequality if the right hand side is $> r + s - \dim(X)$. This implies two things: (a) the induction hypothesis applies to each $Z_j$, and (b) $C \cdot X$ and $\alpha$ intersect properly (because $\alpha$ is a linear combination of those $[W_{i, a_i}^t]$ and $[W_{i, a_i}^t]$ which intersect $Z$ properly). Next, pick $C' \subset \mathbf{P}^N \times \mathbf{P}^1$ as in Lemma \ref{lemma-move} with respect to $C$, $X$, and $W_{i, a_i}^t$, $W_{i, b_i}^t$. Write $C' \cdot X \times \mathbf{P}^1 = \sum n_k [E_k]$ for some subvarieties $E_k \subset X \times \mathbf{P}^1$ of dimension $s + 1$. Note that $n_k > 0$ for all $k$ by Proposition \ref{proposition-positivity}. By Lemma \ref{lemma-transfer} we have $$ [Z] + \sum m_j [Z_j] = \sum n_k[E_{k, 0}]_s $$ Since $E_{k, 0} \subset C \cap X$ we see that $[E_{k, 0}]_s$ and $\alpha$ intersect properly. On the other hand, the cycle $$ \gamma = \sum n_k[E_{k, \infty}]_s $$ is supported on $C'_\infty \cap X$ and hence properly intersects each $W_{i, a_i}^t$, $W_{i, b_i}^t$. Thus by the base case and linearity, we see that $$ \gamma \cdot \alpha \sim_{rat} 0 $$ As we have seen that $E_{k, 0}$ and $E_{k, \infty}$ intersect $\alpha$ properly Lemma \ref{lemma-well-defined-special-case} applied to $E_k \subset X \times \mathbf{P}^1$ and $\alpha$ gives $$ [E_{k, 0}] \cdot \alpha \sim_{rat} [E_{k, \infty}] \cdot \alpha $$ Putting everything together we have \begin{align*} [Z] \cdot \alpha & = (\sum n_k[E_{k, 0}]_r - \sum m_j[Z_j]) \cdot \alpha \\ & \sim_{rat} \sum n_k [E_{k, 0}] \cdot \alpha \quad (\text{by induction hypothesis})\\ & \sim_{rat} \sum n_k [E_{k, \infty}] \cdot \alpha \quad (\text{by the lemma})\\ & = \gamma \cdot \alpha \\ & \sim_{rat} 0 \quad (\text{by base case}) \end{align*} This finishes the proof.