I'm struggling a bit to prove the Cancellation Law for multiplication for the Natural numbers, using induction.

So far I have this:

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Having $a, b \in N$ e $S = \lbrace c \in N | a \cdot c = b \cdot c \longrightarrow a = b \rbrace$

We know that $1 \in S$, since $a \cdot 1 = a = b = b \cdot 1$, 0 is not the base case because even if $a \neq b$, $a \cdot 0 = 0 = b \cdot 0$.

Using $k \in S$, by definition $a \cdot k = b \cdot k \longrightarrow a=b$, we have to show that $k' \in S$

Assuming that $a \cdot k' = b \cdot k'$, we have $a \cdot k + a = b \cdot k + b$, and I don't know where to go from here.

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I tried using something like [this](http://www.fen.bilkent.edu.tr/~franz/nt/nthw01a.pdf), however I'm still stuck.

Thanks for any help