I define $F$ to be the set of functions such that if $f \in F$ then $\lim_{x \rightarrow a} f(x) = f(a)$. Given that we can prove the theorem in the following way:
> $2^3 = 8$ proves that $\lim_{x \rightarrow 2} x^3 = 8$ since $x^3 \in F$.
To see that $x^3 \in F$ I can prove that $x \in F$ and that if $f, g \in F$ then $fg \in F$.