diff --git a/src/op/system/two.tex b/src/op/system/two.tex
index 95d9af6..2be55ba 100644
--- a/src/op/system/two.tex
+++ b/src/op/system/two.tex
@@ -261,7 +261,7 @@
     Thus $\dpn{P^{-1/2}\xi, P^{-1/2}\xi}{H} \ge \dpn{(P+\eps)^{-1/2}\xi, (P+\eps)^{-1/2}\xi}{H}$ for all $\xi \in P^{1/2}(H)$. In particular, since $A(H) \subset P^{1/2}(H)$, 
     \begin{align*}
     \dpn{P^{-1/2}A\xi, P^{-1/2}A\xi}{H} &\ge \dpn{(P+\eps)^{-1/2}A\xi, (P+\eps)^{-1/2}A\xi}{H} \\
-    \dpn{(P^{-1/2}A)^*(P^{1/2}A) \xi, \xi}{H} &\ge \dpn{A^*(P+\eps)^{-1}A\xi,\xi}{H}
+    \dpn{(P^{-1/2}A)^*(P^{-1/2}A) \xi, \xi}{H} &\ge \dpn{A^*(P+\eps)^{-1}A\xi,\xi}{H}
     \end{align*}
     
     so
@@ -310,7 +310,7 @@
 
     and
     \begin{align*}
-        \dpn{(A + A^*)\xi, \xi}{H} &= 2\text{Re}\dpn{A + A^*\xi, \xi}{H} \\
+        \dpn{(A + A^*)\xi, \xi}{H} &= 2\text{Re}\dpn{(A + A^*)\xi, \xi}{H} \\
         |\dpn{(A + A^*)\xi, \xi}{H}| &\le 2 \dpn{P\xi, \xi}{H}^{1/2}\dpn{Q\xi, \xi}{H}^{1/2} \\
         \dpn{(P + A + A^* + Q)\xi, \xi}{H} &\le \dpn{P\xi, \xi}{H} + 2 \dpn{P\xi, \xi}{H}^{1/2}\dpn{Q\xi, \xi}{H}^{1/2} + \dpn{Q\xi, \xi}{H} \\
         &= (\dpn{P\xi, \xi}{H}^{1/2} + \dpn{Q\xi, \xi}{H}^{1/2})^2 \\