Confused about time dilain length contraction

[teodorakis]

hi, we all know when we derive the time dialtion formula we choose a vertical light clock, and basically since the light take a longer zig-zag path to the stationary observer, the time is dilated. That's ok but when deriving length contraction, firstly we accept time dilation for horizontal clock and then do some maths and derive the length contraction formula, so the horizontal clocks affected by both time dilation and length contraction, but what i want to ask is, does length contraction occurs as a result of time dilation?
I mean think of a vertical and a horizontal clock in a spaceship both in sync in that reference frame, now to an observer that has a speed v according to that ship these two clocks must dilate at the same rate by the postulate of relativity, they are not in sync though. Anyway i can understand they should all dely at the same rate, cause otherwise we have a detector of absolute motion, so no matter the choice of clock we use they should be all sync in an inertial reference frame(if we sync them of course). But when i want to mathematically want to derive the time dilaion in horizontal clock i have to accept the length contraction at first, i can't derive it alone, or like all of the sources did, time is dilated same rate as vertical clock adn from here derive thelegth contraction formula. I'm like in a vicious circle, can not derive one ithout other matheatically, please help me.

 

[Tantalos]

In relativistic theory all physics laws are the same independent of the reference frame. When time dilates, so must length also change in order that the speed is preserved: speed = length / time.

 

[teodorakis]

let me clarify things, we don't need to know the machinery or principle of the other clocks, they have to delay asme rate as the vertical light clock, otherwisse the observer in the moving vehicle has a way of understanding that he is moving by using the mismatch in his clocks.
That's ok, but i am trying to derive the time dilation formula in the horizontal clock, just like we derive in vertical clcok, such that i only accept that the speed of light is constant for all observers, with just this assumption, i think we should be able to derive the time dilation alone, shouldn't we?