Let $X\subseteq\mathbb{P}_{K}^{N}$ be an irreducible projective variety where $K$ is an algebraically closed field. Let $\text{dim}(X)=d$. As in Chapter I of Hartshorne's book, we may define $\text{deg}(X)$ as $d!$ times the first term of the Hilbert polynomial of $X$. But there is another definition in for example Wikipedia, saying that degree equals to the intersection number with a general linear space of codimension $d$. I try to formulate this statement as follows:
For hyperplanes $(H_1,\dots,H_d)$ in an open dense subset of $((\mathbb{P}_{K}^{N})^{\vee})^d$, $X\cap H_1\cap\dots\cap H_d$ is reduced and contains exactly $\text{deg}(X)$ closed points.
Could you please help me figure this out? I guess this might be a Bertini type result. Indeed, using appropriate Bertini theorems, I'm convinced that the result is true if we choose the general hyperplanes sequentially, i.e. the general hyperplane $H_i$ depends on $H_1,\dots,H_{i-1}$. But it seems that this is weaker than the statement above. Thank you!