Richard - please - did I give not a valid general proof?

Ok, here - I guess/hope - comes a proof. If this turns out to be correct and useful, then all credits have to be given to John Hendrickson, because his remark gave me the crucial hint.

The postulate looks like this:

Using Johns relation we can rewrite the denominator of right hand side to

$$x+y-\text{Min}(x,y) = \frac{\text{Min}(x,y) \text{Max}(x,y)}{\text{Max}(x,y)-\text{Abs}(x-y)}$$

and here the right hand side simplifies to $\text{Max}(x,y)$:

FullSimplify[(Min[x, y] Max[x, y]/(Max[x, y] - Abs[x - y])) == 
  Max[x, y], {x, y} \[Element] Reals]

so now the postulate looks like so:

and what is left to do is simply to show that

FullSimplify[Max[x, y] - Min[x, y] == Abs[x - y], {x, y} \[Element] Reals]

which evaluates to True.

I think I see what you are getting at and it is interesting use. However my intuition is that proving the below is the same as proving one term of the sums above. Therefore I suggest to prove in two steps. One step such as the below, and another step to prove that any step is not unique.

In:     Abs[x - y]/Max[x, y] + Min[x, y]/(x + y - Min[x, y]) // FullSimplify
Out:    1

I'm not familiar with a way to move your statement as x[[i]] (a random sequence of Z) and for Simplify[] to apply rules to it within a Sum[]. I feel the statistical library could do it. I believe it can be proven by Probability as "the sum of two complementary events equals 1", that is, P(E) + P(E') = 1, where each P is N(E)/N(S) in which E is the count of simple events and S is the total subspace.

Richard, to get rid of the error message you should use Indexed instead of Part:

postulate = Sum[Abs[Indexed[x, {i}] - Indexed[y, {i}]], {i, 1, p}]/Sum[Max[Indexed[x, {i}], Indexed[y, {i}]], {i, 1, p}] == 
 1 - Sum[Min[Indexed[x, {i}], Indexed[y, {i}]], {i, 1, p}]/(Sum[Indexed[x, {i}], {i, 1, p}] + 
    Sum[Indexed[y, {i}], {i, 1, p}] - Sum[Min[Indexed[x, {i}], Indexed[y, {i}]], {i, 1, p}])

But I do not know how to prove this remarkable formula, unfortunately. Empirical checks always evaluate to True, e.g.:

dim = 50;
postulate /. {p -> dim, x -> RandomInteger[{-5, 5}, dim], y -> RandomInteger[{-5, 5}, dim]}

... well, not very helpful, I know!