Intersection of halfspaces and hyperplanes

Question

If $H_1$, $H_2$ and $H_3$ are hyperplanes in an $n$ dimensional vector space $V$ then I want to prove that the linear span of $ H_1\cap H_2^+\cap H_3$ is $H_1\cap H_3$. Clearly the span is contained in $H_1 \cap H_3$. But I could not prove the equality. I tried to prove that both sides have same dimension. But I couldn’t find that the dimension of $H_1 \cap H_2$ is exactly $n-2$. Please help me.