  A version of the classical Klee-And\^o Theorem states the following: For
every Banach space $X$, ordered by a closed generating cone $C\subseteq X$,
there exists some $\alpha>0$ so that, for every $x\in X$, there exist
$x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and
$\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$.
  The conclusion of the Klee-And\^o Theorem is what is known as a conormality
property.
  We prove stronger and somewhat more general versions of the Klee-And\^o
Theorem for both conormality and coadditivity (a property that is intimately
related to conormality). A corollary to our result shows that the functions
$x\mapsto x^{\pm}$, as above, may be chosen to be bounded, continuous, and
positively homogeneous, with a similar conclusion yielded for coadditivity.
Furthermore, we show that the Klee-And\^o Theorem generalizes beyond ordered
Banach spaces to Banach spaces endowed with arbitrary collections of cones.
Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theorem
for cone-valued multi-functions/correspondences.
  We very briefly discuss a potential further strengthening of The Klee-And\^o
Theorem beyond what is proven in this paper, and motivate a conjecture that
there exists a Banach space $X$, ordered by a closed generating cone
$C\subseteq X$, for which there exist no Lipschitz functions
$(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$.
