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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.