Other examples of such many-proof theorems include:

1) Harmonic series diverges

There are at least 8 proofs of this fact based on different ideas.
The first four of them can be found under this link: https://proofwiki.org/wiki/Harmonic_Series_is_Divergent

The other four are of the following form:
"If a subseries diverges, then the whole series diverges too. Now let's prove that the subseries of harmonic series, that formed only by reciprocals of primes diverges..."

And there are at least four different ways to prove it: https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes

2) Variance of a random variable with support in $[0; 1]$ does not exceed $\frac{1}{4}$

There are at least three proofs of this, based on different ideas: What is the largest possible variance of a random variable on $[0; 1]$?

3) No group is both a direct product and a direct product of non-trivial groups

There are at least three proofs of this, based on different ideas: Does there exist a group that is both a free product and a direct product of nontrivial groups?

4) $(\mathbb{Q}, +)$ is not a direct product of any two non-trivial groups

There are at least three proofs of this, based on different ideas: Is $(\mathbb{Q},+)$ the direct product of two non-trivial subgroups?