# Proof 8.17

## Suppose f and g are isometries. Prove that the composition is also an isometry.

Proof: Suppose and are isometries which means they are onto, one-to-one, and distance-preserving. Define the composition of and as . A function is said to be onto if there is a point, x, in the domain for which where any is an element of the codomain. In the relation to the composition, , there needs to be a point in the domain for which where is in the codomain. Since we know that is an isometry, we know that is onto and any point maps to a point . We also know that is onto, so any point in the domain maps to the codomain. Therefore, the composition of and is onto. A function is said to be one-to-one if implies . Pick two points and . If , then it is implied that because is one-to-one by definition. Notice if , because is one-to-one by definition. Since , we can conclude that the composition is one-to-one. A function is said to be distance-preserving if the distance between to points and is the same as the distance between the images, and . Pick points and . Notice, because is distance preserving by definition. Also notice, because is also distance preserving by definition. Since we know these two things, we can conclude that . Therefore, the composition of and is distance-preserving. Since is onto, one-to-one, and distance-preserving, we can conclude that the composition of and is an isometry if and are isometries.