Left Cancellable Element is Left Cancellable in Subset

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

Let $\left ({T, \circ}\right) \subseteq \left ({S, \circ}\right)$.

Let $x \in T$ be left cancellable in $S$.

Then $x$ is also left cancellable in $T$.

Proof
Let $x \in T$ be left cancellable in $S$.

That is:
 * $\forall a, b \in S: x \circ a = x \circ b \implies a = b$

Therefore:
 * $\forall c, d \in T: x \circ c = x \circ d \implies c = d$

Thus $x$ is left cancellable in $T$.