If `A sube B and C sube D`, then prove that:
`A xx C sube B xx D`

Explanation

To prove that if \( A \subseteq B \) and \( C \subseteq D \), then \( A \times C \subseteq B \times D \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- We know that \( A \subseteq B \) means every element of \( A \) is also an element of \( B \).
- Similarly, \( C \subseteq D \) means every element of \( C \) is also an element of \( D \).
- The Cartesian product \( A \times C \) consists of all ordered pairs \( (a, c) \) where \( a \in A \) and \( c \in C \).

2. **Let \( (a, c) \) be an arbitrary element of \( A \times C \)**:
- By definition of the Cartesian product, if \( (a, c) \in A \times C \), then \( a \in A \) and \( c \in C \).

3. **Using the subset relations**:
- Since \( A \subseteq B \), it follows that \( a \in B \).
- Since \( C \subseteq D \), it follows that \( c \in D \).

4. **Forming the ordered pair**:
- From the above, we can conclude that \( (a, c) \) can be expressed as \( (a, c) \in B \times D \) because \( a \in B \) and \( c \in D \).

5. **Conclusion**:
- Since \( (a, c) \) was an arbitrary element of \( A \times C \), and we have shown that it is also in \( B \times D \), we conclude that:
\[ A \times C \subseteq B \times D \]

### Final Statement:
Thus, we have proved that if \( A \subseteq B \) and \( C \subseteq D \), then \( A \times C \subseteq B \times D \).

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