To prove this equation, we need to simplify the left side (LHS) of the equation and show that it is equal to the right side (RHS).

First, let's simplify the LHS:

Given: |x-1|^(lg^2 (x) - lg(x^2))

Since lg(x^2) = 2lg(x), the expression becomes:

|x-1|^(lg^2 (x) - 2lg(x))

Applying the properties of logarithms (a^b = e^(b*ln(a))), we get:

|x-1|^(lg^2 (x) - 2lg(x)) = |x-1|^(lg(x) * (lg(x) - 2))

Since lg(x) - 2 = lg(x) - lg(4) = lg(x/4), the expression becomes:

|x-1|^(lg(x) * lg(x/4))

Using the property of absolute value (|a| = |-a|), we can rewrite the expression:

|x-1|^(lg(x) * lg(4/x))

Now, we know that lg(a) * lg(1/a) = 0 for any positive real number "a". Therefore,

lg(x) * lg(4/x) = 0

Thus, the expression simplifies to:

|x-1|^0 = 1

Therefore, the left side simplifies to 1.

Now, let's look at the right side (RHS) of the equation:

|x-1|^3

Therefore, we have shown that the LHS equals the RHS, and the given equation is proved.