# Diagonal Relation is Universally Compatible

## Theorem

The diagonal relation $\Delta_S$ on a set $S$ is universally compatible with every operation on $S$.

## Proof

Let $\struct {S, \circ}$ be any algebraic structure.

 $\ds$  $\ds x_1 \Delta_S x_2 \land y_1 \Delta_S y_2$ $\ds$ $\leadsto$ $\ds x_1 = x_2 \land y_1 = y_2$ Definition of Diagonal Relation $\ds$ $\leadsto$ $\ds x_1 \circ y_1 = x_2 \circ y_2$ (consequence of equality) $\ds$ $\leadsto$ $\ds \paren {x_1 \circ y_1} \mathrel {\Delta_S} \paren {x_2 \circ y_2}$ Definition of Diagonal Relation

$\blacksquare$