Define a real number $\alpha$ to be computable if there is a computable total function $f_\alpha$ that, given any rational $\epsilon$, yields a rational within $\epsilon$-vicinity of $\alpha$.
Now, assume some principal universal function $U = U(n, x)$ for the class of computable functions. This $U$ generates a certain numbering of computable real numbers: for each computable $\alpha$, its corresponding number is any $n$ such that $U_n = f_\alpha$ (note that every $\alpha$ has infintely many assigned numbers as per Rice's theorem).
Given that, how does one prove the existence of an algorithm that, given any two numbers $n, m$ assigned to any two computable reals $\alpha, \beta$, produces some number that's assigned to their sum $\alpha + \beta$?
So this is a sketch of a proof that came to my mind after writing down this question.
It can be shown that there exists a computable bijection between $\mathbb{N}^2$ and $\mathbb{N}$, so let's denote $[i, j]$ for the natural that corresponds to the pair $(i, j)$ in that bijection. Now define a binary function $F$ such that $F([i, j], \epsilon) = U(i, \epsilon/2) + U(j, \epsilon/2)$. Note that $F$ is clearly computable, and if $i, j$ are indices of some computable reals, then $F_{[i, j]}$ is the function corresponding to their sum.
Since $U$ is principal and $F$ is computable, there exists a computable total $s_F$ such that $\forall n, x : F(n, x) = U(s_F(n), x)$. Combining that with the above, we get that $A(i, j) = s_F([i, j])$ is precisely the algorithm that, given two indices of computable reals, produces an index of their sum.
Does it sound reasonable?