Lemma 7.4.4.label Let $A \subset \real^{2}$ be an algebra. If

1. (S)

   There exists $(x, y) \in A$ such that $x \ne y$.

then $A$ is one of the following:

1. (1)

   $\real^{2}$.

2. (2)

   $\text{span}\bracs{(0, 1)}$.

3. (3)

   $\text{span}\bracs{(1, 0)}$.