Can someone verify that my proof is correct?

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

If $\displaystyle AB=A_1B_1$ then exists $\displaystyle I(A) = A_1 \wedge I(B) = B_1 $ from definition of isometry $\displaystyle 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 $\displaystyle p$ for example point $\displaystyle C$ and prove it that if $\displaystyle I(C) = C_1 $ then $\displaystyle AC=A_1C_1$, so we can conclude that there is $\displaystyle I(p)=p_1$.