If $ mk = nk $ and $ k \neq 0 $, then $ m = n $.
I try to do it by induction. Let $ X = \{k \in \omega \ \wedge \ k \neq 0 \mid mk = nk \Rightarrow n = k \} $
Clearly $ 1 \in X $. Suppose $ k \in X $. To show that $ k ^ + \in X $, I assume that $ mk ^ + = nk ^ + $, by definition of multiplication it follows that $ mk + m = nk + n $, but since I use the induction hypothesis there to get to $ m = n $ ?.