For instance, how does the proof for 7 being prime work?
We can start from 1 and work up to to 7 and show that 7 has exactly two factors, namely 1 and 7. But, how do we rigorously establish that no number greater than 7 can be a factor of 7?
The definition of factor is as follows: For all n, For all x element of N, x is a factor of n iff There exists k element of N in such a way that n=kx
So, the question can be rephrased as a proof that For all n, For all x element of N, x is greater than n => For all k element of N, n is not equal to kx