Hi! Im here to show off just a single example of two axioms and a few theorems derived from them.
So, our alphabet is {A,B,C,a,b,c,1,0}, and our axiom set A is {ABC,CABCA,abABC10,AbaBCc}
First, lets see if ABC is true or not. It is false, since its a substring of both CABCA and abABC10.