Understanding the introduction of a symbol in Einstein's paper In Einstein's paper "On the Electrodynamics of Moving bodies" (1905); first he introduces two sets of coordinates, for two inertial frame moving with relative velocity of $v$: ($x$, $y$, $z$, $t$) and ($\xi$, $\eta$, $\zeta$, $\tau$).
Then he introduces one more coordinate, $x'$, as $x' = x - vt$. 
A possible explanation is: Because of Galilean transformation $x = \xi + vt$, so that $x - vt$ is constant wrt to time in the moving frame k for a particle stationary in k. But is it fine to assume the Galilean transformations in order to derive that Galilean transformations must be replaced by Lorentz transformations?
 A: I believe you are getting things wrong. One thing is to talk about transformation of measurements between reference frames, the other is to talk about the description of the motion of a reference frame with respect to the other.
In the first situation we have two sets of measurements $(t,x,y,z)$ and $(\tau,\xi,\eta,\zeta)$ with the meaning that we should understand the first tuple being what an observer on the origin of the first frame measures and the second tuple the same thing for an observer on the origin of the second frame.
In the second situation we have a single observer: the one on the first frame. He sees a second observer on the origin of the second frame and follows its motion. He will end up with a sequence of events $(t,x,y,z)$, comprising the worldline of said observer in the origin of the second frame, i.e., all events in his existence.
In particular, suppose we have the following situation: when $t = 0$ for the first observer on the first reference frame, he and the other observer stand together with their sets of axes paralel. Then the second reference frame (and hence the second observer) starts moving along the common $x,\xi$ axis with uniform velocity $v$.
Forget measurements on the second frame. If the first observer sees the second reference frame and hence the second observer moving with velocity $v$ he will conclude that at his time $t$ the coordinates of the second observer are $(t,vt,0,0)$.
He can then define $x'$ to be
$$x'=x-vt$$
and this is just the distance seen by the first observer on the first reference frame of the point in space $(x,0,0)$ to the instantaneous origin of the second frame $(vt,0,0)$. This isn't a galilean transformation yet. 
There is no claim whatsoever here that $x' = \xi$, i.e., that $x'$ matches the distance to the same material point in space to the origin of the second observer as seen by him. 
Indeed, we have lenght contraction. Let now $\xi'$ be the the $\xi$ coordinate of the same point, as seen now by the second observer. It can be characterized as: the distance of said point to the origin of the second frame, as seen in the second frame.
So $x'$ and $\xi'$ are distances between the same material points, but as seen by two different reference frames, one moving with respect to the other with velocity $v$. Then Lorentz contraction yields
$$x'=\dfrac{1}{\gamma}\xi'.$$
Or also $\xi' = \gamma x'$ which means that $$\xi' = \gamma (x -vt)$$ which is the correct Lorentz transformation law.
So: one is not using Galilean transformations to derive Lorentz transformations. One is only observing that the measurement of coordiantes can be thought of as measurements of distances on each frame and computing these distances and relating then by Lorentz contraction.
The thing is that once you postulate that the speed of light is the same in all reference frames, you must give up on the idea of absolute simultaniety of events. This in turn, as pointed out by Einstein, forces the idea of relative time on us. These ideas leads directly to time dilation and lenght contraction. This is what makes the above discussion lead to the Lorentz transformations instead of the Galilean ones.
If there was no absolute speed, which takes the same values for every observer and hence one did not abandon absolute simultaniety of events as above, we would conclude that the distances are the same for the two reference frames, and only on this case we would conclude $x'=\xi'$ and hence $\xi' = x-vt$ which is the Galilean transformation law.
A: It seems to be a Galilean transformation between two coordinate systems with relative velocity v in x direction. 
The Galilean transformations form a part of the symmetry group of Newtonian mechanics. 

But is it fine to assume the Galilean transformations in order to derive that Galilean transformations must be replaced by Lorentz transformations? 

Because of, for example, the Michelson-Morley-Experiment, Einstein, like a few other Physicists of that time, came to the conclusion, that the Newtonian concept of absolute space and time isn't true and therefore that the Galilean Transformations had to be replaced by some other ones.  
If you consider non-relativistic speeds, it is fine to assume the Galilean Transformations, since they proved to be the right thing when dealing with velocities $v<<c$. 
But they don't work with the first postulate of Special Relativity, namely that the speed of light in the vacuum is the same for all oberservers (as it was indicated by the Michelson-Morley-Experiment). Einstein derived the Lorentz Transformations based on his two postulates of Special Relativity and the fact that they had to


*

*reduce to the Galilean Transformations for low speeds

*be linear because the Galilean Transformations are.

