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The question you provided asks for the assumption and proof in a direct proof of the theorem:

**Theorem:** For any two integers \( x \) and \( y \), if \( x + y \) is even, then \( x \) and \( y \) have the same parity (i.e., both are either even or both are odd).

Let's analyze the options:

- **Option A:** 
  - **Assumed:** Integers \( x \) and \( y \) are both even or both odd.
  - **Proven:** \( x + y \) is even.
  
  This assumption and proof are the reverse of the theorem. It assumes the conclusion (same parity) and then proves that the sum is even, which is not the direction needed in the proof of the theorem.

- **Option B:**
  - **Assumed:** \( x \) and \( y \) are integers such that \( x + y \) is even.
  - **Proven:** \( x \) and \( y \) are both even.
  
  This doesn't fully capture the theorem. It suggests that the proof only works for even numbers, which is incomplete because the theorem applies whether both numbers are even or both are odd.

- **Option C:** 
  - **Assumed:** \( x \) and \( y \) are integers such that \( x + y \) is even.
  - **Proven:** \( x \) and \( y \) are both even or both odd.
  
  This option correctly assumes that the sum is even and then proves the actual theorem: that both \( x \) and \( y \) have the same parity (either both are even or both are odd). This is the correct option.

- **Option D:** 
  - **Assumed:** \( x \) and \( y \) are even integers.
  - **Proven:** \( x + y \) is even.
  
  This is similar to Option A, as it assumes one specific case (both \( x \) and \( y \) are even) and proves the sum is even. This is not what the theorem states, as it does not include the case where both are odd.

### Correct Answer: **C**

Would you like a deeper explanation of the theorem or the process? Feel free to ask.

Here are 5 questions to consider:
1. Why is the assumption that \( x + y \) is even important in this proof?
2. How do you prove that if \( x + y \) is even, then \( x \) and \( y \) have the same parity?
3. Can this theorem be extended to more than two integers?
4. What is meant by "parity" in the context of integers?
5. How does the direct proof method differ from other proof techniques like contradiction?

**Tip:** To prove parity theorems, use the definitions of even and odd numbers, and manipulate them algebraically to show their relationships.

Theorem: For any two integers, x and y, if x + y is even, then x and y have the same parity. Which facts are assumed and which facts are proven in a direct proof of the theorem?

Explore how the sum of two integers leads to the same parity (either both are even or both are odd) in this detailed proof. Learn how to identify assumptions and conclusions, with step-by-step reasoning.

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