Affine subspaces proposition

Question

I try to prove that a subset of an affine space is an affine space iff it contains the line through every two distinct points of it.

Answer 1

This is essentially the definition:

A space $A$ if affine iff for all $x,y \in A$ we have $\lambda x+ (1-\lambda )y \in A$ for all $\lambda \in \mathbb{R}$.