Proof Structures

Given a statement to prove, the proper way to start off the proof is to assume the variables that are assigned the "for all" quantifier are true.
Then for the following variables (say x) assigned the existential quantifier, you have to find an actual example that belongs in the set that satisfies the conditions.
It is sometimes very difficult to find an example that will fit into the proof, so it may be easier work backwards.
That's why in the proof, we say variable x is equal to, in the beginning, an unknown element.

We start at the implication and rearrange the antecedent in such a way that resembles to what we are trying to prove in the consequent.
Most of the time, x can be easily found from the scratch work done from the process of rearranging the antecedent.

Going back to the proof structure, we now know what x is, and so now we can finish the full proof with an x that works, and that will complete the proof.