In practice, how to clearly prove that variables are uncorrelated? Suppose that there is a property $\mathbf{p}$, that for each study case $i$ we want to estimate as: $$p_i = c_1 x_{i1}+c_2 x_{i2}$$
If $p_i$ and $x_{i2}$ are uncorrelated, then we simply would do
$$p_i = c^\prime _1 x_{i1}$$
In practice: How to be sure that this procedure is correct?
Wikipedia article on Uncorrelated random variables states that two random uncorrelated variables $X$ and $Y$ satisfy $$0= E(XY)-E(X) E(Y)$$
but I bet it won't be the case for a finite number of observations.
¿Can (in practice) it be used to say something about the correlation between $p_i$ and $x_{i2}$?
Alternatively, I think that the corresponding p-value from the bivariate regression can be enough to state that: it is very probably that there is not correlation and because of that the first model is not better than the second one. Is this correct? Are there other useful, convenient or necessary test that I can use for the purpose?
Edit: I found that the two answers are complementary and equally useful and interesting. So in few days I will choose the less voted answer to try to equally the recognition that they receive.
 A: If the assumption that the regression errors are i.i.d and normal is correct  then yes, the $p$ value is precisely what you want to look at. It tells you that, to keep your notation, $x_2$ is linearly uncorrelated to $p$ after controlling for $x_1$.
However a few issues might arise. 
First, you might not believe that the errors are normal. In this case you should generate confidence intervals, and $p$ values, through re-sampling bootstrap to make sure.
Second, even if there is no linear correlation there might be some non-linear effect. You want to capture that effect to improve your prediction of $p$. The easiest way to spot a non-linear correlation while keeping safely within the linear regression confines is probably the Frisch-Waugh-Lovell theorem which gives you a nice visual clue about possible structure. I gave a simple R example here.
A: The following statement (paraphrased your question a bit) is not quite true:
Given that $p_i$ and $x_{i2}$ are uncorrelated, the model $p_i \sim x_{i1} + x_{i2}$ can be reduced to $p_i \sim x_{i1}$.

In practice: How to be sure that this procedure is correct?

You have to define "correctness."  One definition of correctness could be -- "The reduced model have lower mean squared error than the original model", -- in which case it's easy to show a counter example in which the statement need not be true.
You can check a discussion and example here: https://stats.stackexchange.com/a/206663/54725, in which I constructively show an example where a variable (here $x_{i2}$) that is uncorrelated with the outcome (here $p_i$) can actually have a statistically-significant non-zero coefficient in the regression model and improve predictive accuracy of the model.  Note that this is true even if we are only considering linear relationships between variables, and even if you ignore all artifacts due to finite data.
To answer your original question: How do I prove that two variables $X$ and $Y$ are uncorrelated?


*

*If you have reasons to believe that $X$ and $Y$ are drawn from a multivariate normal distribution, then you can test whether the Pearson product-moment correlation is 0 at some significance level.

*If you don't know the relationship between $X$ and $Y$, then this problem can be hard.


*

*If $X$ and $Y$ are nominal, then you can use the Chi-squared test of independence: https://onlinecourses.science.psu.edu/stat500/node/56.

*If $X$ and $Y$ are real valued, you could also try to use distance correlation (https://en.wikipedia.org/wiki/Distance_correlation), which can help establish a stronger property that two variables are independent of each other (and are hence uncorrelated).  I don't know off-hand any statistical test that uses this correlation, but you could always bootstrap the distribution of the sample's distance correlation and work with it.

*I don't know how to deal with cases that are a mix of categorical and continuous variables.  Perhaps someone else can add a pointer!
Hope this helps.
