In "Elliptic Curves: Number Theory and Cryptography" we see in the proof of proposition 2.21 that for the separable endomorphism $\alpha : E(\overline{K}) \rightarrow E(\overline{K})$ for every point $(a,b)$ with $(a,b) \neq \infty$ and $a,b \neq 0$ there are exactly $\deg(\alpha)$ points with $\alpha(x_1,y_1) = (a,b)$.
How can we then conclude that the kernel also has $\deg(\alpha)$ points since we excluded $\infty$? Do we look at the limit as $(a,b)$ tends towards $\infty$?