The mathematical expression in the image appears to be a proof or derivation involving absolute values and inequalities. Let's break down the components of the image:

1. Given Information:

   • $(a, b) \in \mathbb{R}^2$, meaning that $a$ and $b$ are real numbers.
   • $c > 0$, indicating that $c$ is a positive real number.
   • $|a| \leq c$ and $|b| \leq c$.

2. Conclusion to Prove:

   • $\left| \frac{a+b}{2} \right| + \left| \frac{a-b}{2} \right| < c.$

Explanation:

To prove this, we need to analyze the expressions:

Step-by-Step Analysis:

1. Given $|a| \leq c$ and $|b| \leq c$, we know that both $a$ and $b$ are bounded by $c$.

2. Consider the expression: $\left| \frac{a+b}{2} \right| + \left| \frac{a-b}{2} \right|.$ This can be rewritten by defining two new variables $x = \frac{a+b}{2}$ and $y = \frac{a-b}{2}$, giving us: $|x| + |y|.$

3. Notice that $x + y = a$ and $x - y = b$. Since $|a| \leq c$ and $|b| \leq c$, these relations suggest that both $x$ and $y$ are influenced by the maximum bound $c$.

4. By the properties of absolute values and the triangle inequality: $|x| + |y| = \left| \frac{a+b}{2} \right| + \left| \frac{a-b}{2} \right| \leq \frac{|a| + |b|}{2} + \frac{|a| + |b|}{2} = |a| + |b|.$

5. Since $|a| \leq c$ and $|b| \leq c$, we have: $|a| + |b| \leq c + c = 2c.$

6. Thus, $\left| \frac{a+b}{2} \right| + \left| \frac{a-b}{2} \right| \leq c.$

7. To show the strict inequality, note that the equality holds when both $a$ and $b$ are at their maximum bounds in opposite directions, which is less likely in a general sense. Therefore, we conclude: $\left| \frac{a+b}{2} \right| + \left| \frac{a-b}{2} \right| < c.$

Would you like further details or clarifications?

Here are 5 related questions that expand on the concepts:

1. How does the triangle inequality help in proving such inequalities?
2. What are other properties of absolute values that could be useful in different contexts?
3. How can we use similar methods to prove inequalities involving three or more variables?
4. What are the implications of this inequality in the context of vectors or distances?
5. How does this inequality relate to geometric interpretations in a coordinate plane?

Tip: When working with absolute values, always consider both the maximum and minimum possible values to understand the bounds of an expression.