Definition: A real algebraic variety in $\mathbb{R}^n$ is the set of common real zeroes of some $f_1, \dots , f_s \in \mathbb{R}[x_1, \dots , x_n]$.
By the dimension of a real algebraic variety $V$ we mean its dimension as a manifold, or alternatively it is the maximal integer $d$ such that there is a homeomorphism of $[0,1]^d$ into $V$. The reason for this definition is that for example with the normal definition of the dimension of a variety, the variety $(x^2+y^2)$ would have dimension 1, but the set of real zeroes consists of a single point so we would like to say it has dimension 0.
Question: Is it true that if $V \subset \mathbb{R}^n$ is an irreducible non-degenerate real algebraic variety of dimension $d \ge 2$, then there exists some hyperplane $H$ in $\mathbb{R}^n$ such that $H \cap V$ is non-degenerate, irreducible and of dimension $d-1$? If so, how would one prove existence?
For complex varieties, you can prove similar statements using the Theorem of Bertini and/or Lefschetz. But what about for real varieties?