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# Direct proof

A **direct proof** is a way of showing that something is true or false by using logic. This is done by combining known facts. No assumptions are made when doing a direct proof. Lemmas and theorems are used to prove direct proofs.

A statement that can be proved with a direct proof is usually in the form "if *p*, then *q*." Here, *p* and *q* are facts. To solve a statement like this, every case where *p* is true must be considered.

For example, this statement can be solved with a direct proof: "if x and y are even integers, then x+y is an even integer." Since x and y are even, then we can say that *x=2m* and *y=2n,* where *m* and *n* are integers. (This is a lemma.) We can also say that *m+n* is an integer, because adding two integers gives an integer. (This is another lemma.) Then we can say that *x+y=2m+2n=2(m+n)*. Since *m+n* is an integer, we can say *2(m+n)=2k*, where *k=m+n*. We know that any integer times 2 is an even integer. We can then say that any two even integers added together give an even integer.

Direct proofs are used in mathematics, logic, and computer science. The opposite of a direct proof is an indirect proof (also called a *proof by contradiction*.)