Can someone verify that my proof is correct?

I have to prove that if A,B are points of line p and A_1, B_1 are points of line p_1 and if AB=A_1B_1, then there is isometry I which transforms line p into line p_1 and also is I(A)=A_1, I(B)=B_1.

If AB=A_1B_1 then exists I(A) = A_1  \wedge I(B) = B_1 from definition of isometry I(A) = A_1  \wedge I(B) = B_1  \Rightarrow AB = A_1 B_1 .

Knowing that isometry preserves congruence, colinearity (hope grammar term is correct) and order of points then we can chose any point on line p for example point C and prove it that if I(C) = C_1 then AC=A_1C_1, so we can conclude that there is I(p)=p_1.