# Left-cancellative

## Definition 2

Let $$(S, \circ)$$ be an algebraic structure. Then $$\circ$$ is left-cancellative if and only if

$$\forall a,b,c \in S: a \circ b = a \circ c \Rightarrow b = c$$
THEOREM

$$\forall a,b,c \in \mathbb{N} : a + b = a + c \Rightarrow b = c$$

Proof available

LEMMA

$$\forall x,y,z \in \mathbb{Z} : z + x = z + y \Rightarrow x = y$$