Before revision:  Further, given another sublinear increasing functional \mathcal{E}(\cdot)\colon\mathcal{L}( U \times V )\to\mathbb{R} which leaves the constants invariant, the tower property \mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot| U )) is characterized via a pasting property of the representing sets of probabilities .
After revision:  Further, given another sublinear increasing functional \mathcal{E}(\cdot)\colon\mathcal{L}( U \times V )\to\mathbb{R} which leaves the constants invariant, the tower property \mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot| \mathcal{G )) is characterized via a pasting property of the representing sets of probabilities .