math

1 files changed, 7 insertions, 8 deletions

diff --git a/proofs.tex b/proofs.tex
index 2cff729..585e134 100644
--- a/proofs.tex
+++ b/proofs.tex
@@ -316,14 +316,13 @@ In particular,
 \begin{gather*}
         \forall S\subseteq\mathcal{N},\quad A(S)^{-1} \succeq A(S\cup\{i\})^{-1}
 \end{gather*}
-as $A(S)\leq A(S\cup\{i\})$. Observe that
-    \begin{gather*}
-        \forall S\subseteq\mathcal{N}\setminus\{i\},\quad
-        P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\geq
-        P_{\mathcal{N}\setminus\{i\}}^\lambda(S\cup\{i\}),\\
-        \forall S\subseteq\mathcal{N},\quad P_{\mathcal{N}\setminus\{i\}}^\lambda(S) 
-        \geq P_\mathcal{N}^\lambda(S).
-    \end{gather*}
+as $A(S)\preceq A(S\cup\{i\})$. Observe that
+ %   \begin{gather*}
+ %       \forall 
+$P_{\mathcal{N}\setminus\{i\}}^\lambda(S)\geq P_{\mathcal{N}\setminus\{i\}}^\lambda(S\cup\{i\})$ for all $S\subseteq\mathcal{N}\setminus\{i\}$, and
+       % ,\\
+        $P_{\mathcal{N}\setminus\{i\}}^\lambda(S) \geq P_\mathcal{N}^\lambda(S),$ for all $S\subseteq\mathcal{N}$.
+ %\end{gather*}
     Hence,
     \begin{align*}
         \partial_i F(\lambda)