Lipschitz continuity of matrix product

Question

Let $x\in\mathbb{R}^n$ and define the matrix functions $A:\mathbb{R}^n\to\mathbb{R}^{m\times p}$ and $B:\mathbb{R}^{n}\to\mathbb{R}^{p\times p}$. Define $F(x) = A(x)\left[B(x)\right]^{-1}A(x)^\top\in\mathbb{R}^{m\times m}$. I am interesting in showing that $F(x)$ is Lipschitz continuous, that is, I want to show that there exists some $L_F>0$ such that, \begin{align*} \|F(x) - F(y)\|\le L_F\|x-y\| \end{align*} Is showing that $A(x)$ and $B(x)$ are bounded functions of $x$ and Lipschitz with respect to some (any, because of norm equivalence) matrix norm sufficient for showing that $F(x)$ is Lipschitz?