# Cancellation Laws/Proof 3

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## Corollary to Group has Latin Square Property

Let $G$ be a group.

Let $a, b, c \in G$.

Then the following hold:

- Right cancellation law

- $b a = c a \implies b = c$

- Left cancellation law

- $a b = a c \implies b = c$

## Proof

Suppose $x = b a = c a$.

By Group has Latin Square Property, there exists exactly one $y \in G$ such that $x = y a$.

That is, $x = b a = c a \implies b = c$.

Similarly, suppose $x = a b = a c$.

Again by Group has Latin Square Property, there exists exactly one $y \in G$ such that $x = a y$.

That is, $a b = a c \implies b = c$.

$\blacksquare$

## Sources

- 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.4$: Theorem $1$ Corollary