q) is true, we assume that p is
true and use that to show that q must also be true.4. To prove that an if--then statement (p q) is true, we assume that p is
true and use that to show that q must also be true.
Proof Suppose that a = d + rm and b = d + sm for some integers d, r, and s.
| Subtract these two equations to obtain: a - b | = | (d + rm) - (d + sm) |
| a - b | = | rm - sm |
| a | = | b + (r - s)m |
| a | = | b + tm where t = r - s |
| Add c to both sides of the equation: a + c | = | b + c + tm |
Since r and s are both integers, r + s will be an integer. Therefore a + c = b + c + tm for some integer t.