The image shows a proof involving partial derivatives and summation, likely in the context of minimizing a function, possibly related to least-squares estimation or regression analysis.

The proof seems to involve:

1. Taking the derivative of a function with respect to a parameter, likely aiming to find a minimum or maximum.
2. Setting the derivative equal to zero to find critical points.
3. The expression $\sum (y - \beta_0 - \beta_1 x) = 0$ suggests a focus on the residuals $(y - \beta_0 - \beta_1 x)$, common in regression.

Here's a breakdown of the steps observed:

1. Define Objective: Likely minimizing the sum of squared residuals, $\sum (y_i - \beta_0 - \beta_1 x_i)^2$.
2. Differentiate with respect to $\beta_0$: The partial derivative with respect to $\beta_0$ is set to zero, yielding $\sum (y - \beta_0 - \beta_1 x) = 0$.
3. Interpretation of Results: This step is usually used to find the best-fit parameters in regression analysis.

Would you like a detailed explanation of the steps involved, or do you have any specific questions about the proof? Here are some related questions to consider:

1. What is the goal of minimizing the sum of squared residuals in regression?
2. How does taking partial derivatives help in finding the best-fit line?
3. Why set the derivative equal to zero in optimization problems?
4. What role do $\beta_0$ and $\beta_1$ play in a linear regression model?
5. How does this approach generalize to multiple regression with more than one predictor?

Tip: When working with proofs in statistics or regression, focusing on the role of each term in the objective function can clarify the logic behind each step.