First of all, I should note that I'm working on Peano's axioms for constructing natural numbers.
For example, lets try to prove that "every natural number is the sum of four squares" by induction.The proof as follows;
i-) Show that $0$ satisfies the condition.
ii-) Assume that $n$ also satisfies the condition, and derive that $n+1$ also satisfies the condition.
The induction should work for proving this kind of a theorem because 0 will satisfies the condition and by applying $ii-)$ where $n=0$, $1$ will also satisfy, and repeating the same procedure, all the natural numbers will satisfy the condition.
However, lets consider the theorem;
"Every non-zero element of N has blah."
In the proof of this kind of a theorem, what it has been done is that they show that $0$ satisfies the theorem (since 0 makes the assumption false), and again apply the second condition of induction.
But why this is also work as it works in the first example ? What is the logic ?
I should also note that, induction is given as the fourth axiom of Peano, so I cannot talk about the proof of induction, but what I am asking is an intuition for the logic why it works.