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 H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), “The Capacity Region of the Gaussian Multiple-Input Multiple-Output Broadcast Channel,” IEEE Trans. IT, 52 (9): 3936-3964 (2006). Now, assume that case 2 holds. Following, the above proof for case 1, one can get = 1: In this case, equation (22) implies that I(V ; U jZ) = 0. Furthermore equation (21) will hold with equality. Since t(x) 2 T , we must have I(U ; Y ) = I(U ; Z) = 0. The fact that I(V ; U jZ) = I(U ; Y ) = I(U ; Z) = 0 implies that I(U ; Y ) = I(U ; ZV ) = 0. Therefore the inequality I(U ; Y ) I(U ; ZV ) also holds in this case.
Let us consider the following two cases: < 1: In this case, equation (22) implies that I(U ; Y ) I(U ; V Z) + 1 I(V ; U jZ). This inequality implies the desired inequality I(U ; Y ) I(U ; V Z).