I'm asked to prove as a corollary to: the image of the line segment PQ under F is a line segment between F(P) and F(Q), that an isometry preserves straight lines.
Here is the proof I've written down:
Let F be an isometry.
Let F(P) be denoted by P'.
Let P,Q be arbitrary points on a line L. Let X be a point on the line segment PQ.
Since F preserves distances we know that: d(P,X)=d(P',X') and d(Q,X)=d(Q',X').
We have d(P,Q)=d(P,X)+d(X,Q).
By assumption of F preserving distance we also have d(P',Q')=d(P',X')+d(X',Q').
Thus the image of segment PQ is contained in its image, segment P'Q'; but since P,Q are arbitrary, the segment PQ may be considered indefinitely long. Thus the line L is mapped to a straight line L', since the arbitrarily large segments composing that line are straight.
Is it correct? Am I writing too much? Is there a better way? Am I missing a second part? ie. to prove to sets equal you must show each belongs to the other.