Answer: To prove that f(x) + r is also a coordinate system when l is a line in absolute geometry and f is a coordinate system, we need to show that it satisfies the three properties of a coordinate system: uniqueness, order-preservation, and segment determination.
Uniqueness: Each point on the line l is associated with a unique real number in the coordinate system f(x) + r. This means that for every point P on the line l, there is only one real number x such that f(x) + r corresponds to P.
Order-preservation: If point P is to the left of point Q on the line l, then the corresponding real number x for P is less than the corresponding real number x for Q. In other words, if P and Q are two points on the line l such that P is to the left of Q, then f(x_P) + r < f(x_Q) + r.
Segment determination: For any two points P and Q on the line l, if R is a point between P and Q, then the corresponding real number x_R for R is between the corresponding real numbers x_P and x_Q. In other words, if P, Q, and R are three points on the line l such that R is between P and Q, then f(x_P) + r < f(x_R) + r < f(x_Q) + r.
These properties hold for the coordinate system f(x) + r, as adding a constant r to f(x) does not affect the uniqueness, order-preservation, or segment determination of the coordinate system.
Therefore, we have shown that f(x) + r is also a coordinate system for the line l in absolute geometry.