**Proof: a × c is an improper subset of b × c**

To prove that a × c is an improper subset of b × c, we need to show that every element in a × c is also an element of b × c.

Let's assume that a is an improper subset of b. This means that every element in a is also an element of b, but b may have additional elements that are not in a.

Now, let's consider an arbitrary element (x, y) in a × c. Here, x represents an element from set a and y represents an element from set c. We need to show that this element is also an element of b × c.

**Element (x, y) in a × c**

Since (x, y) is an element of a × c, it means that x is an element of a and y is an element of c.

**Proof: (x, y) is an element of b × c**

To show that (x, y) is an element of b × c, we need to demonstrate that x is an element of b and y is an element of c.

Since a is an improper subset of b, we know that x is an element of a, and therefore x is also an element of b.

Similarly, since y is an element of c, it follows that y is an element of c.

Hence, (x, y) is an element of b × c.

**Conclusion: a × c is an improper subset of b × c**

Since we have shown that every element in a × c is also an element of b × c, we can conclude that a × c is an improper subset of b × c.

This proof relies on the assumption that a is an improper subset of b. If this assumption is not true, the conclusion may not hold. Therefore, it's important to verify that a is indeed an improper subset of b before using this proof.

An improper subset is a subset containing every element of the original set. A proper subset contains some but not all of the elements of the original set. For example, consider a set {1,2,3,4,5,6}. Then {1,2,4} and {1} are the proper subset while {1,2,3,4,5} is an improper subset.