Left Cancellable Element is Left 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 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 $T$.
$\blacksquare$