= Solution
The <real projective plane> is the set of one-dimensional <linear subspaces> of $\mathbb R^3$. Equivalently,
$$
\boxed{\mathbb{RP}^2=(\mathbb R^3\setminus\{0\})/\sim,\qquad v\sim\lambda v\ (\lambda\in\mathbb R\setminus\{0\}).}
$$
A <projective point> is written in <homogeneous coordinates> $[x:y:z]$. A <projective line> is the set of <projective points> whose representatives belong to one fixed two-dimensional <linear subspace> of $\mathbb R^3$. Such a <linear subspace> is the <kernel of a linear map> of a nonzero <linear functional>, so its equation has the form $\alpha x+\beta y+\gamma z=0$ with $(\alpha,\beta,\gamma)\ne0$.
Two distinct <projective points> have <linearly independent> representatives $p,q$. Their <linear span> has dimension two and hence defines a <projective line> containing both. Conversely, any two-dimensional <linear subspace> containing both representatives contains their entire <linear span> and therefore equals it. Thus \b[exactly one <projective line> passes through two distinct <projective points>].
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