Diffeomorphism between $S^n \backslash B_r(x)$ and some ball $B$ for arbitrary $x \in S^n$.

Question

I was going through my lecture notes and in a proof I have written down that

"... Since we have a diffeomorphism between $S^n \backslash B^{n+1}_r(x)$ and some ball for arbitary $x \in S^n$ ..."

and I was wondering about said diffeomorphism. Could anybody help me?

Thanks in advance!