Suppose there is a dominant morphism $H: \mathbb{A}^1_k \times W \to X$ such that $H(0, -) \neq H(1, -)$ as morphisms from $W$ to $X$. Here $W$ and $X$ are varieties over a field $k$. Assume that $dim(W) > dim(X) - 1$, then there is a hypersurface $W^{\prime}$ in $W$ such that the restriction of $H: \mathbb{A}^1_k \times W^{\prime} \to X$ is still dominant.
I am trying to reduce the dimension of fibre in the restriction. But I can't properly get it. Please help by giving hint. Thanks in advance.