The problem is written in Greek, and it translates to:

"If $x$ and $y$ are two real numbers, prove that if $xy = 0$, then $x = 0$ or $y = 0$."

Proof:

To prove that if $xy = 0$, then either $x = 0$ or $y = 0$, we can use the following reasoning:

1. Given: $x$ and $y$ are real numbers, and $xy = 0$.

2. If the product of two real numbers is zero, at least one of the numbers must be zero. This follows from a basic property of real numbers, known as the zero-product property.

   • Assume $xy = 0$.

   • If $x \neq 0$, then we can divide both sides of the equation $xy = 0$ by $x$. This gives: $y = \frac{0}{x} = 0.$

   • Alternatively, if $y \neq 0$, then we can divide both sides of the equation $xy = 0$ by $y$. This gives: $x = \frac{0}{y} = 0.$

3. Therefore, if $xy = 0$, then either $x = 0$ or $y = 0$.

This completes the proof.

Would you like more details or have any questions? Here are some related questions:

1. What is the zero-product property, and why is it important in algebra?
2. Can you give examples of situations where this property is used in equations?
3. How would this property apply to complex numbers or matrices?
4. Is the zero-product property valid for all types of numbers (like real, complex, etc.)?
5. How does this concept relate to solving quadratic equations?

Tip: Understanding the zero-product property is crucial for solving various algebraic equations, especially quadratic ones.