The (revised) question to answer:
Can anyone give an example of a serious proof using this funny (revised) theorem?
For any natural number $n$ and prime $p<n-1$ there exist a prime $q$ so that $n\equiv p\pmod q$
(The theorem has a very easy proof).
With a serious proof one could mean a proof where the theorem is needed for the construction and the idea of the proof.
The question might seem awkward, but arise from a wonder about mathematics in a future when computers might generate millions of maybe easy-proved but somewhat non intuitive theorems. Could such theorems, which not are the result of the intuition of mathematicians, be of any importance?