In this subsection, "location" is meant in the general sense, i.e. not as point-like location only.

**(Theorem)** If l is a straight line in K parallel to **v**, and l' is the corresponding location in K' marked out at (coordinate) time t in K, then l' is a straight line parallel to **v**'.

**(Proof)** Viewed from K', any point P of l is moving at the constant velocity **v**'. On the other hand, in K all points of l', and only those from K', go through P. Thus, during the course of its movement in K', P meets exactly the points of l'. So l' is a straight line parallel to **v**'.

**(Theorem)** The location l' marked out in K' is the same for every t, and l' corresponds to l at any time t' in K'.

**(Proof)** The points of K' marked out at any t_{1} in K, and only those, are moving along l in K, so marking out at any other t_{2} in K results in the exact same points of K'. Now, switching the roles of K and K', we get that for every t' in K', l' corresponds in K to the very same straight line l_{l'} parallel to **v**. But from the way l' was constructed we also know that l_{l'} and l have common points, so l_{l'} must be equal to l, since both of them are parallel to **v**.

The x'-axis of K' will always be chosen such that it coincides with the x-axis of K, at all times. To obtain the x'-axis, all we have to do is to mark out the location in K' that corresponds to the x-axis at any given time t in K. As for the orientation of the x'-axis, we define the order of its points, and with that the positive and the negative direction as well, by the order of the corresponding points on the x-axis at t. Independently of our choice of t, we'll get the same x'-axis and the same order of its points.

**(Theorem)** In K', **v**' points in the negative direction of the x'-axis.

**(Proof)** In K, a point P of the x-axis meets the points of the x'-axis in the above defined negative direction. If we imagine there is a particle at P, then causality implies that P, while moving at velocity **v**', meets the points of the x'-axis in the same order in K' (and so the above defined "directions" are proper directions in K' too).

For completeness, one last theorem about the x'-axis:

**(Theorem)** If in K', point P_{2}' comes after P_{1}' on the x'-axis, then at any given t', the corresponding point P_{2} comes after P_{1} on the x-axis of K.

**(Proof)** Both P_{1} and P_{2} are moving in the negative direction along the x'-axis in K'. At t', P_{1} is already at P_{1}', so P_{2}' must have met P_{1} before t'. Therefore, P_{2}' meets in K' first P_{1} then P_{2}. Due to causality, the order is the same in K. And since P_{2}' is moving in the positive direction along the x-axis in K, P_{2} comes after P_{1}.

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