Alternative proofs of Schwartz–Zippel lemma I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz. 

Are there any other proofs which use substantially different ideas?
As a followup to Per Vognsen's answer, Dana Moshkovitz's proof already suggests a really easy proof for only a slightly weaker version of the Schwartz-Zippel Lemma that, I think, suffices for most applications. 

Let $f : \mathbb{F}^n \to \mathbb{F}$ be a nonzero polynomial of degree $d$, where $\mathbb{F}$ is a finite field of order $q$, and let $x \in \mathbb{F}^n$ be a point such that $f(x) \neq 0$.  There are $(q^n-1)/(q-1)$ many distinct lines passing through $x$ such that they partition $\mathbb{F}^n-\{x\}$.  The restriction of $f$ to each of these lines is a degree $d$ univariate polynomial, which is nonzero, because it is nonzero at $x$, and so, has at most $d$ zeros.  Thus, the total number of zeros of $f$ is at most $d(q^n-1)/(q-1)$.  Schwartz-Zippel, for comparison, gives the stronger upper bound of $d q^{n-1}$.

Given this proof's easiness, I'm sure it's folklore; if not, it should be :)  I'd appreciate if someone could provide a reference.