Let $M,N\subset \mathbb R^p$ be manifolds of class $C^1$ such that $\dim M+\dim N\lt p$. I want to prove the sets of the points $x-y\in \mathbb R^p$ such that $x\in M$ and $y\in N$ has measure zero.
My attempt
I'm trying to use the theorem below:
Sard Theorem: Let $M$ and $N$ be two manifolds with dimension $m$ and $f:M\to N$ a function of class $C^1$. If $S$ is the set of points $x\in M$ such that the derivative $f'(x):T_xM\to T_{f(x)}N$ is not an isomorphism, then $f(S)$ has measure zero in $N$.
So I have to prove the determinant of $f'(x-y):T_{x-y}M\to T_{f(x-y)}N$ is zero, am I on the right direction?