In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_1,...,A_m\subseteq {ℝ}^n$ be subsets with finite Lebesgue measure. Then, for any sequence $f_0,...,f_m$ of $ℝ$-linearly independent polynomials in the polynomial ring $ℝ[X_1,...,X_n]$ there are real numbers ${\lambda }_0,...,{\lambda }_m$, not all zero, such that the real affine variety $\{x\in {ℝ}^n;{\lambda }_0{f}_0\left(x\right)+\cdots +{\lambda }_m{f}_m\left(x\right)=0\}$ simultaneously bisects each of subsets $A_k$, $k=1,...,m$. Then some its applications are studied.