I recently set about studying the basics of set theory prior to my first year as a Mathmajor in Colombia and I've been proving properties of sets but I've perceived the greater part of them can be proven in the same way. Then, instead of doing so I was trying to prove each in a different way: by ...

Assume that two balls $B_1,B_2$ of radius $r$ continuously move around inside of a square of size $d$. They bounce off the walls, i.e. the $x$-component of the velocity is multiplied with $-1$ when they hit the left/right wall, and similarly for the $y$-component. Apart from that, the velocity is...