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The question asks about the theorem: 

**"For any two real numbers, \( x \) and \( y \), if \( x \) and \( y \) are both rational, then \( x + y \) is also rational."** 

The task is to identify which assumptions and proven facts match this theorem.

Let's break down the options:

- **Option A:**
  - Assumed: \( x \) is rational and \( y \) is rational.
  - Proven: \( x + y \) is rational.
  - This matches the theorem exactly, where we assume that both \( x \) and \( y \) are rational, and we are proving that their sum is rational.

- **Option B:**
  - Assumed: \( x + y \) is rational.
  - Proven: \( x \) is rational or \( y \) is rational.
  - This is not correct because the theorem is proving something about the sum of two rational numbers, not making assumptions about the sum first.

- **Option C:**
  - Assumed: \( x \) is rational or \( y \) is rational.
  - Proven: \( x + y \) is rational.
  - This is incorrect because the theorem requires both \( x \) and \( y \) to be rational to guarantee their sum is rational.

- **Option D:**
  - Assumed: \( x + y \) is irrational.
  - Proven: \( x \) is irrational and \( y \) is irrational.
  - This is not related to the theorem at all, as the theorem deals with sums of rational numbers, not irrational numbers.

**Correct Answer: A**

Would you like more details on the reasoning? Here are some related questions:

1. What defines a rational number mathematically?
2. Can the sum of a rational number and an irrational number ever be rational?
3. What is the difference between a direct and an indirect proof?
4. How can you prove that the sum of two rational numbers is always rational?
5. Why is option B an incorrect assumption about the sum of rational numbers?

**Tip:** When proving properties of numbers, always check the definitions (like rational or irrational) and see if they hold under operations such as addition or multiplication.

For any two real numbers, x and y, if x and y are both rational, then x + y is also rational. Which facts are assumed, and which facts are proven in a direct proof of the theorem?

This problem explains how the sum of two rational numbers is also rational, illustrating key concepts of real numbers and proof strategies. It includes assumptions and conclusions in mathematical theorems.

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