Proof me wrong! - The order when establishing proof

An important part of complying with scientific standards is the use of the scientific method to determine whether an equation holds more than empirical truth in itself. What we need here is a method for understanding, properly, if a mathematical expression holds truth. We will focus on the form this time around.

If we were to use the example:

∀ x ∈ Z, | x | > 0 ⇒ ∃ z ∈ Z, x < z

We would have the following steps for establishing the proof:

        Assume ∀ x ∈ Z # domain assumption 

                Assume | x | > 0 # antecedent 

                        Let z' = ... # some value that depends on x 

                        [proof of z' ∈ Z] 

                        Then, z' ∈ Z 

                        [proof of x < z']

                        Then, x < z' 

                Then ∃ z ∈ Z, x < z # introducing ∃
        Then ∀ x ∈ Z, | x | > 0 ⇒ ∃ z ∈ Z, x < z

Let's dissect these steps, separating it by their indentation. We would have, in this case, 3 indents:

• Overall proof
• If 'antecedent', then 'right-hand side statement'
• Proof of 'right-hand side statement' for limited or universal variable

If we were to work it the other way around, we would have:

• Proof that the 'right-hand side statement' holds itself true for a variable that belongs to the desired domain and scope (∃ or ∀)
• Since it was proven that the 'right-hand side statement' is true for a variable that belongs to the domain and scope, given the antecedent, we can conclude that it holds true for the desired domain
• Since it was proven with the antecedent, and the antecedent depends on the domain limitations, then the whole statement must be true.

While this was explained using several words, it is important not only to learn the structure, but also to understand why it is used in such a way.