  We deal with the problem of coexistence in interval effect algebras using the
notion of a witness mapping. Suppose that we are given an interval effect
algebra $E$, a coexistent subset $S$ of $E$, a witness mapping $\beta$ for $S$,
and an element $t\in E\setminus S$. We study the question whether there is a
witness mapping $\beta_t$ for $S\cup\{t\}$ such that $\beta_t$ is an extension
of $\beta$. In the main result, we prove that such an extension exists if and
only if there is a mapping $e_t$ from finite subsets of $S$ to $E$ satisfying
certain conditions. The main result is then applied several times to prove
claims of the type "If $t$ has a such-and-such relationship to $S$ and $\beta$,
then $\beta_t$ exists".
