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Need a mathematical proof of a statement

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Project Description

Generate a random number xi at each time from time 1 to time N, then you got a random number array X~{x1, x2,...xn} belonging to a N dimensional space;
Then generate a random number yi at each time from time 1 to time N, then you got a random number array Y~{y1, y2,...yn} belonging to a N dimensional space;
Let Z=f(X,Y)

Now, generate a random number array X'~{x'1, x'2,...x'n} at time T, belonging to a N dimensional space;
Then generate a random number array Y'~{y'1, y'2,...y'n} at time T, belonging to a N dimensional space;
Let Z'=f(X',Y')

Prove Z and Z' are from the same distribution. Or at least, the randomness of Z', X' and Y' are the same as Z, X and Y even though they are generated/calculated differently.

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