Question: I want to show that this *magic* equation is true for all kinds numbers (rational, irrational, imaginary, natural, etc..) which is true as far as I've tested.$$\displaystyle\large\frac{2x+10}{2}-x=5$$How would I exactly do that?

Answer: 
First, we have

$$\frac{2x + 10}{2} = \frac{2(x + 5)}{2} = x + 5$$

so the overall equation simplifies to

$$x + 5 - x = 5$$

which is obviously true for all $x$.
$$\frac{2x + 10}{2} - x = \frac{2x}{2} + \frac{10}{2} - x = x + 5 - x = 5$$