Proof:

To prove the given statement, we need to show that:

x^3 * y^3 * z^3 - 3xyz = (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)

Expanding the left side:

x^3 * y^3 * z^3 - 3xyz

= (x * y * z) * (x^2 * y^2 * z^2) - (x * y * z) * 3

= (x * y * z) * (x^2 * y^2 * z^2 - 3)

Expanding the right side:

(x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)

= (x * y * z) * (x^2 * y^2 * z^2) - (x * y * z) * (xy + xz + yz)

= (x * y * z) * (x^2 * y^2 * z^2) - (x^2 * y * z^2 + x * y^2 * z^2 + x * y * z^2)

Comparing both sides:

We can see that the expanded forms of both sides are the same:

(x * y * z) * (x^2 * y^2 * z^2 - 3) = (x * y * z) * (x^2 * y^2 * z^2) - (x^2 * y * z^2 + x * y^2 * z^2 + x * y * z^2)

Since both sides of the equation are equal, we can conclude that:

x^3 * y^3 * z^3 - 3xyz = (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)

Therefore, the given statement is proved.

Summary:

By expanding and comparing the left and right sides of the equation, we have shown that x^3 * y^3 * z^3 - 3xyz is equal to (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz). Hence, the given statement is proved.