Condition for Operation to be Left Distributive over Constant Operation

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

Let $\sqbrk c$ be the constant operation for some $c \in S$.

Then:
 * $\circ$ is left distributive over $\sqbrk c$


 * $\forall x \in S: x \circ c = c$
 * $\forall x \in S: x \circ c = c$

Sufficient Condition
Let $\circ$ be left distributive over $\sqbrk c$.

That is:
 * $\forall x \in S: x \circ c = c$

Necessary Condition
Let:
 * $\forall x \in S: x \circ c = c$

Then:

That is, $\circ$ is left distributive over $\sqbrk c$.