Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation

Theorem
Let $\R^2$ denote the cartesian plane.

Let $\alpha$ denote the relation defined on $\R^2$ by:
 * $\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$

The equivalence classes of $\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$.

Proof
We have from Cross-Relation on Real Numbers is Equivalence Relation that $\alpha$ is an equivalence relation.

Thus:

Thus each equivalence classes consists of sets of points such that:


 * $x - y = c$

That is:
 * $y = x + c$

Thus from Equation of Straight Line in Plane, this is the equation of a straight line whose slope is $1$.