Injection iff Left Cancellable/Necessary Condition

Theorem
Let $f: Y \to Z$ be an injection.

Then $f$ is left cancellable.

Proof
From the definition: a mapping $f: Y \to Z$ is left cancellable :


 * $\forall X: \forall g_1: X \to Y, g_2: X \to Y: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$

Let $f: Y \to Z$ be an injection.

Let $X$ be a set

Let $g_1: X \to Y, g_2: X \to Y$ be mappings such that:
 * $f \circ g_1 = f \circ g_2$

Then $\forall x \in X$:

As $f$ is an injection, $g_1 \left({x}\right) = g_2 \left({x}\right)$ and thus the condition for left cancellability holds.