Proving statements regarding bases of affine spaces

! Definitions: Basis of an affine subspace;

Consider a $$d$$-dimensional affine subset $$D$$ of an affine space, and a set of points $$S$$ in $$D$$.

• If $$S$$ is a generating set for $$D$$, then $$S$$ contains at least $$d+1$$ points. A generating set with exactly $$d+1$$ points is a basis.

• If $$S$$ is an independent set for $$D$$, then $$S$$ contains at most $$d+1$$ points. An independent set with exactly $$d+1$$ points is a basis.

• If $$|S| = d + 1$$ and $$S$$ is not contained in a proper affine subspace of $$D$$, then $$S$$ is a basis.

Let's take the first bullet for example:

$$\dim(D) = d$$, meaning that $$D$$ is generated by $$d+1$$ (independent) points. Suppose that $$S$$ is a generating set for $$D$$. This means that the points in $$S$$ are not contained in a proper affine subspace of $$D$$. From this fact follows that we cannot find an affine subspace of $$D$$ that is generated by strictly less than $$d+1$$ points, which also holds for $$S$$. This results in $$|S| \ge d+1$$. I am struggling to find an explanation for the second part of the statement.

Is my attempt correct?

I'll give my proofs for the other two bullets too:

• Suppose that $$S$$ is an independent set for $$D$$. By definition, for every point $$x \in S$$ there exists an affine subspace of $$D$$ that contains $$S \backslash \{x\}$$. Suppose that $$|S| > d+1$$ and take an arbitrary point $$s \in S$$. There has to be an affine subspace of $$D$$ that contains $$S \backslash \{s\}$$. However $$|S \backslash \{s\}| > d$$, meaning that there cannot be an affine subspace of $$D$$ (dimension $$d$$) that contains $$S \backslash \{s\}$$. This results in $$|S| \le d+1$$.

• This one would follow directly from the definition of a generating set of an affine space and the first bullet.

Thank you!