The second step is to gather together a list of related definitions, axioms and theorems that have already been proven. This will help you write a rigorous proof because it will give you a list of exact statements that can be used as justifications. Be sure to include all definitions of concepts stated in the statement you are trying to prove. Also, include the definitions of anything used in the sketch you made in step one.

For a constructive proof you proceed from the hypothesis, using their definitions or related axioms and write down a sequence of true statements (each justified by one of the definitions, axioms or theorems gathered in step 2). The statements you choose to make will depend on the ideas you had in step 1. Sometimes, if you've convinced yourself using a diagram, you can go through the steps used in drawing the picture and write corresponding statements. For example, if you drew a line through two points you can quote the first axiom of geometry. Then if you drew a point between them on the line, you can use a betweeneess axiom and so on.

If an axiom or statement tells you that one of two possibilities is true: statement A or statement B, then you need to proceed with each statement as a seperate case. You can only use one of the statements in each case. Sometimes one statement will lead to a contradiction of some axiom or hypothesis, in which case the other case must be true.

If you decide to do a proof by contradiction then you either contradict the entire statement you are trying to prove or just the conclusion. Then you proceed to find out what would happen if that contradictory statement were true using axioms and definitions etc. Usually the pictures won't make much sense in a proof by contradiction since the essence of the proof is that something is wrong and leads to a contradiction. Nevertheless, it is often easier to do a proof by contradiction than a constructive proof because you need only lead to a contradictory statement and not towards a specific conclusive statement.

When writing a proof by contradiction you must be careful to write a correct contradiction of the statement. "All cows are brown and white" is contradicted by "there exists a cow which is not brown or not white". "There exists a purple cow" is contradicted by "all cows are not purple" which includes the possibility that there are no cows.

When proving an if and only if statement, break the proof into a forwards proof and a backwards proof. A iff B must be proven first A implies B and then B implies A. Similarly a proof that two sets are equal requires a forwards and backwards proof, first proving a point in the first set must be in the second set and then the opposite. Don't forget both parts of these proofs.

When you have finished writing a proof (which can take a long time!), you should check it. Go through the whole proof and verify that every statement made in the proof is justified with a precise reference. If something isn't justified then you should try to figure out if you think it is true. If you think its true but needs a few lines to prove, then call it a Lemma and prove it on another page.