q) is true, we assume that p is
true and show that q is also true.3. To prove that an if--then statement (p q) is true, we assume that p is
true and show that q is also true.
Proof Suppose that a and b are integers and a = 4x + 1 and b = 4y + 1 for some integers x and y.
| Thus a · b | = | (4x + 1)(4y + 1) |
| = | 16xy + 4x + 4y + 1 | |
| = | 4(4xy + x + y) + 1 |
Since x and y are both integers, 4xy + x + y 
Z
by closure of Z. Therefore the product a · b = 4m + 1 for some
integer m (namely m = 4xy + x + y).