In addition to theorems, we also have things called "lemmas." A lemma is a "little theorem" which is used to help prove a bigger theorem. Usually we call something a lemma if we don't really think we're going to need to re-use it, beyond proving the bigger theorem. Let's look at an example. I'm going to introduce a theorem, and I'll call it "Mr. T's Odd Square Theorem."

Mr. T's Odd Square Theorem: An integer is odd if and only if its square is one more than a multiple of four.

This is something new; we've introduced the terminology "if and only if." This means that the statement works both ways:

If an integer is odd, then its square is one more than a multiple of four.
If an integer's square is one more than a multiple of four, then the number is odd.

In order to prove the theorem, we have to prove both statements above. These are our two lemmas.

Lemma One: If an integer is odd, then its square is one more than a multiple of four.

STATEMENT                                                    REASON
1. Let x be any odd number                                   Variable assignment
2. x = 2y + 1, for some integer y                            Definition of odd number
3. x2 = (2y + 1)2                                            Squaring property of equality*
4. x2 = 2y(2y + 1) + 1(2y + 1)                               Distributive property
5. x2 = 4y2 + 2y + 2y + 1                                    Distributive property
6. x2 = 4y2 + 4y + 1                                         Combining like terms
7. x2 = 4(y2 + y) + 1                                        Distributive property
8. ∴ x2 is one more than a multiple of four                  Definition of multiple 

* The squaring property of equality states that the solutions of the new equation contain all solutions of the original equation.

Lemma Two: If an integer's square is one more than a multiple of four, then the number is odd.

This one we will approach as a proof by contradition; we'll assume that it's not true, and arrive at a contradiction.

STATEMENT                                                 REASON
1. Assume n is an even number, and n2 = 4k + 1            Assumption
2. n = 2j, for some integer j                             Definition of even number
3. n2 = 4j2                                               Squaring property of equality
4. 4j2 = 4k + 1                                           Transitive property (1, 3)
5. 4j2 - 4k = 4k + 1 - 4k                                 Subtraction property of equality
6. 4(j2 - k) = 1                                          Combining like terms, Distributive property
7. 1 is a multiple of 4                                   Definition of multiple

Since 1 is not divisible by 4, this is the contradiction we needed, thus proving Lemma Two.

Since Lemma One and Lemma Two are both true, we have proven both sides of the "if and only if," so we have proved the theorem:

Mr. T's Odd Square Theorem: An integer is odd if and only if its square is one more than a multiple of four

In this section, we introduced you to two different concepts: the idea that "if and only if" indicates that the theorem is "reversible," and a lemma is a mini-proof used to help prove a bigger proof. I've seen lemmas described as "pre-proofs." Something you have to prove before you can prove something else!

1.

State the following as two statements: x + 1 = 3, if and only if x = 2

2.

State the following as two statements: x is even if and only if x - 1 is odd.

3.

State the following as two statements: x is a multiple of three if and only if its cube is a multiple of 3.

4.

State the following as two statements: A number is natural if and only if it is a positive integer.