Prove that $A$ is affine iff, for every $a\in A$, $A-\{a\}$ is a subspace of $\mathbb{R}^n$
I know that to show a subspace I must show $\forall x,y\in A-\{a\}, x+y\in A-\{a\}$ and $\forall \alpha\in\mathbb{R}, x\in A-\{a\}, \alpha x\in A-\{a\}$.
And I believe I can work out how to do that part, but I am struggling with the other direction. We know $A-\{a\}$ is a subspace and need to prove $A$ is affine.