Binomial Representation Theorem proof, Baxter and Rennie

I am currently reading Financial Calculus of Baxter and Rennie and have one question regarding proof of Binomial Representation Theorem. In the text, we assume that increments of $N$ martingale are $$\Delta N_i = \Phi_i \Delta S_i + k_i$$ My question is: why do we assume the linear relation here between increments of these two martingales? I couldn't find any justification for that.

• Is it not the case that $\Delta N_i$ and $\Delta S_i$ can take on only two values? – Bob Jansen Jun 10 '18 at 14:49