A general idea in "for all" statements

I have a proposition for the page Proving "for all" statements.

There is a more general approach/view, which generalize induction and the articles in the subsection Prove the result for some cases and deduce it for the rest.

It is the following: If you have to show a statement $A(x)$ it is useful to exploit the structure of the set $X$ .

If there are for example some operations $T_1,...T_k$ acting on $X$ such that you can write every $x \in X$ as $T_{i_1}^{n_1}...T_{i_m}^{n_m}x_0$ with a special point $x_0$ and you can show:

1) $A(x_0)$ is true

2) If you apply one of the transformations $T_i$ to an $y \in X$ where $A(y)$ is true then $A(T_iy)$ remains true.

Then you have shown your statement. (Induction works like this)

Second example: If your set $X$ is partially ordered, such that every nonempty subset $Y \subset X$ has an element $y_0 \in Y$ which is minimal in the following sence: If $y_0$ and $y \in Y$ are comparable, then $y_0 \leq y$ . Then you can show the statement by assuming that the set of counterexamples is not empty and look at one of the minimal counterexamples. If one can show, that there is no minimal counterexample, the proof is complete.