  Suppose $A$ is a pro-C*-algebra. Let $L_{A}(E)$ be the pro-C*-algebra of
adjointable operators on a Hilbert $A$-module $E$ and let $K_{A}(E)$ be the
closed two sided $*$-ideal of all compact operators on $E$. We prove that if
$E$ be a full Hilbert $A$-module, the innerness of derivations on $K_{A}(E)$
implies the innerness of derivations on $L_{A}(E)$. We show that if $A$ is a
commutative pro-C*-algebra and $E$ is a Hilbert $A$-bimodule then every
derivation on $K_{A}(E)$ is zero. Moreover, if $A$ is a commutative
$\sigma$-C*-algebra and $E$ is a Hilbert $A$-bimodule then every derivation on
$L_{A}(E)$ is zero, too.
