# Diagonal Relation is Universally Compatible

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## Theorem

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

## Proof

Let $\left({S, \circ}\right)$ be any algebraic structure.

\(\displaystyle \) | \(\) | \(\displaystyle x_1 \Delta_S x_2 \land y_1 \Delta_S y_2\) | |||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle x_1 = x_2 \land y_1 = y_2\) | Definition of Diagonal Relation | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle x_1 \circ y_1 = x_2 \circ y_2\) | (consequence of equality) | ||||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle \left({x_1 \circ y_1}\right) \Delta_S \left({x_2 \circ y_2}\right)\) | Definition of Diagonal Relation |

$\blacksquare$