Introduction:
In this question, we are given a finite-dimensional vector space V and three subspaces U, W, and X of V. We are also given that the product of U and W is equal to the product of U and X. Our task is to prove that W is equal to X.

Proof:

Step 1: Assume that U is a subspace of V. Since U is a subspace, it contains the zero vector, denoted as 0U.

Step 2: Consider an arbitrary vector w in W. Since W is a subspace, it must also contain the zero vector, denoted as 0W.

Step 3: Since UW = UX, it implies that every vector in UW is also in UX, and vice versa.

Step 4: Let's consider an arbitrary vector u in U and an arbitrary vector w in W. Since the product of U and W is equal to the product of U and X, we have:

u * w = u * x, where x is an arbitrary vector in X.

Step 5: Let's consider the vector v = w - x. Since X is a subspace, it contains the zero vector, denoted as 0X. Therefore, we have:

v = w - x = 0X.

Step 6: Now, let's consider the vector u * v. Since U is a subspace, it contains the zero vector, denoted as 0U. Therefore, we have:

u * v = u * (w - x) = u * w - u * x = u * w - u * w = 0U.

Step 7: Since 0U is the zero vector of U, it must also be the zero vector of V. Therefore, we have:

u * v = 0V.

Step 8: Now, let's consider the vector u * v = 0V. Since V is a finite-dimensional vector space, it has a unique zero vector. Therefore, we can conclude that u * v = 0V implies v = 0V.

Step 9: Since v = 0V, we have w - x = 0V, which implies w = x.

Step 10: Therefore, we have proved that if UW = UX, then W = X.

Conclusion:
In this question, we proved that if the product of two subspaces U and W is equal to the product of U and X in a finite-dimensional vector space V, then W is equal to X.