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So... I'm trying to get some simplifications on equations using own defined variables, but the tries I gave didn't result on what I wanted.

The problem: I have 2 variables defined as:


x = (z[1, 2] z[3, 4])/(z[1, 3] z[2, 4])

k = (z[1, 3] z[2, 4])/(z[1, 2] z[2, 3] z[3, 4] z[4, 1])

Where


z[a_,b_] := z[a] - z[b]

And i'm trying the result:


k = 1/(x z[2,3]z[4,1])


My tries:


Simplify[Eliminate[{k == (
    z[1, 3] z[2, 4])/(z[1, 2] z[2, 3] z[3, 4] z[4, 1]), 
   x == (z[1, 2] z[3, 4])/(z[1, 3] z[2, 4])}, {z[1, 2], z[2, 4], 
   z[3, 4], z[1, 3]}]]

and


k /. (z[1, 2] z[3, 4])/(z[1, 3] z[2, 4]) -> x (* into this one I only defined k before*)

I searched and so, but either I got something that I already tried or something incredible hard. So I don't know if I got something wrong or is really hard to get substitutions.

EDIT 1

So, this is just an example of how I would like the program to work not just using to print the answer I want. I want him to simplify the expression using my definitions. In the second try I got him to substitute if was only one term(Like z[1,2]) so I'm start to thinking that way don't substitute in a mathematical way.

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ClearAll[x, k, z]
With[{
   k = (z[1, 3] z[2, 4])/(z[1, 2] z[2, 3] z[3, 4] z[4, 1]),
   x = (z[1, 2] z[3, 4])/(z[1, 3] z[2, 4])
   },
  k x
  ]/x
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