Right Cancellable Element is Right Cancellable in Subset

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Theorem

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

Let $\struct {T, \circ} \subseteq \struct {S, \circ}$.

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


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


Proof

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

That is:

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

Therefore:

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

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

$\blacksquare$