I am reading Don Taylor's book 'The Geometry of Classical Groups' and currently I am trying to understand the affine geometry section. There is a lemma which appears to be a criterion for a bijection between two affine spaces to be an affine isomorphism.
Lemma. Suppose that $\mathcal{A}_1$ and $\mathcal{A}_2$ are affine spaces of the same dimension over division rings $K_1$ and $K_2$, $|K_1| \neq 2$. Suppose that $\varphi: \mathcal{A}_1 \rightarrow \mathcal{A}_2$ is a bijection such that whenever $P,Q,R$ are collinear points of $\mathcal{A}_1$, then $\varphi(P),\varphi(Q),\varphi(R)$ are collinear points of $\mathcal{A}_2$. Then $\varphi$ is an affine isomorphism and $\dim \varphi(\mathcal{B}) = \dim \mathcal{B}$ for every affine subspace $\mathcal{B}$ of $\mathcal{A}$.
What I cannot understand is its proof. Given a subspace $\mathcal{B}$, we choose the subspaces $$\mathcal{B}_0 \subset \mathcal{B}_1 \subset \cdots \subset \mathcal{B}_n = \mathcal{A}$$ so that $\dim \mathcal{B}_i = i$ and $\mathcal{B} = \mathcal{B}_k$ for some $k$. Let $\mathcal{C}_i$ be the smallest subspace of $\mathcal{A}_2$ that contains $\varphi(\mathcal{B}_i)$. Since $|K_1|\neq 2$, we can choose distinct points $P,Q$ of $\mathcal{B}_{i+1}\setminus \mathcal{B}_i$ such that $PQ$ (line) contains a point of $\mathcal{B}_i$. Then every point of $\mathcal{B}_{i+1}$ is either on a line through $P$ and a point of $\mathcal{B}_i$ or on a line through $Q$ and a point of $\mathcal{B}_i$. It follows that $\dim \mathcal{C}_{i+1} \leqslant 1 + \dim \mathcal{C}_i$.
This part I do not understand. Can you please help me to understand first, why is 'Since $|K_1|\neq 2$, we can choose distinct points $P,Q$ of $\mathcal{B}_{i+1}\setminus \mathcal{B}_i$ such that $PQ$ (line) contains a point of $\mathcal{B}_i$' and then 'every point of $\mathcal{B}_{i+1}$ is either on a line through $P$ and a point of $\mathcal{B}_i$ or on a line through $Q$ and a point of $\mathcal{B}_i$'? I have tried to draw pictures, but it was not very helpful.