I am trying to understand an idea presented in McNulty book, matriods a geometric introduction about the new hyperplane arrangement $\mathcal{A}^{''} = \{ H \cap H_x | H \in \mathcal{A}\}$ where $\mathcal{A}$ is an affine arrangement of hyperplanes in $\mathbb R^d$ and $H_x \in \mathcal{A}$ and $x$ is a point $\in M(\mathcal{A})$ and so corresponds to a hyperplane $H_x$ in the arrangement.
The book is saying that $$M(\mathcal{A}^{''}) = M (\mathcal{A})/x$$ i.e., the matroid of the induced hyperplane arrangement in x equal the contraction of the matroid of the hyperplane arrangement.
I am not very sure why this statement is correct, can someone show me any kind of proof of it? I am learning matroids by myself so I am not very good at it.