Proof that $x_1=x_2$ maximizes $f(x_1)+f(x_2)$, when its derivatives are unequal until $x_1=x_2$. Given some function $f: \mathbb{R} \rightarrow \mathbb{R}$. define $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ as

$$g(x_1,x_2)=f(x_1)+f(x_2)$$

Assume that $f$ is decreasing in argument. Define $X$ as the sum of $x_1+x_2$. $X$ is assumed to be some finite constant. Hence, whenever $x_1$ increases, the value of $x_2$ decreases by the same amount. Claim: $g(x_1,x_2)$ is maximized when $x_1=x_2$. I argue this using the following fact (that I know independently to be true):

$$\frac{\partial f(x_1)}{\partial x_1}&gt;\frac{\partial f(x_2)}{\partial x_2} \iff f(x_1)&gt;f(x_2).$$

Which (I believe) states that increasing $x_1$ and decreasing $x_2$ increases  $g(x_1,x_2)$, whenever $f(x_1)&gt;f(x_2)$. Note that $f(x_1)&gt;f(x_2)$ $\iff$ $x_1&lt; x_2$ since $f$ is decreasing in its argument. Hence, $g(x_1,x_2)$ can be increased by transferring value from $x_2$ to $x_1$ until $f(x_1)=f(x_2)$. Conversely, I also know that

$$\frac{\partial f(x_1)}{\partial x_1}&lt;\frac{\partial f(x_2)}{\partial x_2} \iff f(x_1)&lt;f(x_2).$$

And we can increase $g$ by transferring value from $x_1$ to $x_2$ whenever $x_1&gt;x_2$. Since this holds everywhere for $g(x_1,x_2)$ whenever $x_1 \neq x_2$ we can increase $g$ by transferring value between $x_1$ and $x_2$. Hence $g$ is maximized when $x_1=x_2$. Does this argument work?
A thought rather than an answer: 

This looks as if it's closely related to the standard ideas of an optimum on the production-possibility curve in Economics. The proofs usually given in textbooks implicitly assume that the PP curve is convex, so that there's a unique maximum. You may be making a similar assumption here. As written, I don't actually believe the claim you're trying to prove, although I confess I haven't read it carefully -- it's too muddled for my taste.
A thought rather than an answer: 

This looks as if it's closely related to the standard ideas of an optimum on the production-possibility curve in Economics. The proofs usually given in textbooks implicitly assume that the PP curve is convex, so that there's a unique maximum. You may be making a similar assumption here. As written, I don't actually believe the claim you're trying to prove, although I confess I haven't read it carefully -- it's too muddled for my taste.