Can someone verify that my proof is correct?

I have to prove that if are points of line and are points of line and if , then there is isometry which transforms line into line and also is .

If then exists from definition of isometry .

Knowing that isometry preserves congruence, colinearity (hope grammar term is correct) and order of points then we can chose any point on line for example point and prove it that if then , so we can conclude that there is .