Mathematics : How to Prove Things
This section is TOTAL CRAP that needs to be rewritten. Sorry.
The activity called "proving" is always ultimately based on Propositional Logic. The scheme is that there is a disjunction of formulae, like , $X_1 \vee X_2 \vee ... \vee X_n$ where $X$ denotes a formula, and then a "complete proof" shows that according to "axioms" (tautologies) all of the formulae, $X$, are true. Sometimes only one of the formulae, $X$, is shown to be true, because that's enough to show that the whole disjunction, the $X_1 \vee X_2 \vee ... \vee X_n$, is true, and it might be difficult to show that the other formulae within the disjunction have a value of "true".
The formulae, $X$, can be very complex and are usually derived from an original problem, which might, but does not have to be, a propositional formula.
The propositional parts of proofs are automated by "theorem provers" (SAT solvers), which take propositional formula $X$ as an input and try to find, if there exists any set of boolean constants that guarantee the formula $X$ to have a boolean value of "true".
Some Classical Heuristics
Find at Least one Instance of a Contra Example
Theorem (tautology) candidate: "All cats on planet Earth are white."
To show that the theorem candidate is not a tautology, an existence of a single non-white cat must be shown.