diff --git a/Dissertation/ann_first_approximations.tex b/Dissertation/ann_first_approximations.tex
index dd83cd8..a5b0f8c 100644
--- a/Dissertation/ann_first_approximations.tex
+++ b/Dissertation/ann_first_approximations.tex
@@ -1,5 +1,5 @@
 \chapter{ANN first approximations}
-\section{ANN Representations for One-Dimensional Identity and some associated properties}
+\section{ANN Representations for One-Dimensional Identity}
 
 \begin{definition}[One Dimensional Identity Neural Network]\label{7.2.1}
 	We will denote by $\id_d \in \neu$ the neural network satisfying for all $d \in \N$ that:
@@ -621,7 +621,7 @@ This completes the proof.
 %\end{proof}
 \section{$\trp^h$, $\etr^{n,h}$ and Neural Network Approximations For the Trapezoidal Rule.}
 \begin{definition}[The $\trp$ neural network]
-	Let $h \in \R_{\ges 0}$. We define the $\trp^h \in \neu$ neural network as:
+	Let $h \in \lb 0,\infty \rp $. We define the $\trp^h \in \neu$ neural network as:
 	\begin{align}
 		\trp^h \coloneqq \aff_{\lb \frac{h}{2} \: \frac{h}{2}\rb,0}
 	\end{align} 
@@ -640,7 +640,7 @@ This completes the proof.
 	This a straight-forward consequence of Lemma \ref{5.3.1}
 \end{proof}
 \begin{definition}[The $\etr$ neural network]
-	Let $n\in \N$ and $h \in \R_{\ges 0}$. We define the neural network $\etr^{n,h} \in \neu$ as:
+	Let $n\in \N$ and $h \in \lb 0,\infty \rp$. We define the neural network $\etr^{n,h} \in \neu$ as:
 	\begin{align}
 		\etr^{n,h} \coloneqq \aff_{\underbrace{\lb \frac{h}{2} \ h \ h\ ... \ h \ \frac{h}{2}\rb}_{n+1-many},0}
 	\end{align}
@@ -1126,7 +1126,7 @@ We will call the approximant $\max_{i \in \{0,1,\hdots, N\}}\{ f_i\}$, the \text
 	\end{align}
 \end{proof}
 \begin{remark}
- 	We may represent the neural network diagram for $\mxm^d$ as:
+ 	We may represent the neural network diagram for $\mxm^d$ below. 
 \end{remark}
 \begin{figure}[h]
 	\begin{center}
diff --git a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex
index f2b2677..565f6c7 100644
--- a/Dissertation/ann_rep_brownian_motion_monte_carlo.tex
+++ b/Dissertation/ann_rep_brownian_motion_monte_carlo.tex
@@ -866,39 +866,56 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
 	\begin{corollary}\label{cor_ues}
 		Let $N,n,\fn \in \N$, $h,\ve \in \lp 0,\infty\rp$, $q\in\lp 2,\infty\rp$, given $\mathsf{UES}^{N,h,q,\ve}_{n,\mathsf{G}_d, \Omega, \fn} \subsetneq \neu $, it is the case that:
 		\begin{align}
-			\E\left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber
+			\E\left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i} ds \rp \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber
 		\end{align}
 	\end{corollary}
+	
+	
+	
+	
+	
 	\begin{proof}
 		Note that \cite[Corollary~3.8]{hutzenthaler_strong_2021} tells us that:
 		\begin{align}
 			&\E\left| \E \lb \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp \cX^{d,t,x}_{r,\Omega}\rp\rb -\frac{1}{\mathfrak{n}}\lb \sum^{\mathfrak{n}}_{i=1}\lb \exp \lp \int_t^T \alpha_d \circ \mathcal{X}^{d,t,x}_{r,\omega_i}\rp ds \cdot \fu_d^T\lp \mathcal{X}^{d,t,x}_{r,\omega_i}\rp\rb \rb \right|\nonumber \\
 			&\les \frac{\fK_p \sqrt{p-1}}{n^{\frac{1}{2}}} \lp \E \lb \left| \exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \right|\rb \rp
 		\end{align}
-		
-		Note that Taylor's theorem states that:
+		For the purposes of this proof let it be the case that $\fF: [0,T] \rightarrow \R$ is the function represented for all $\ft \in \lb 0,T \rb$ as:
 		\begin{align}
-			\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp = 1 + \int^T_t \alpha_d \circ \cX ^{d,t,x}_{r,\Omega}ds + \frac{1}{2}\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds + \fR_3
+			\ff\lp t\rp = \int^T_{T-t} \alpha_d\circ \cX^{d,t,x}_{r,\Omega} ds
 		\end{align}
-		Where $\fR_3$ is the Lagrange form of the reamainder. Thus $\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot u\lp T,\cX^{d,t,x}_{r,\Omega}\rp$ is rendered as:
+		In which case we have that $\fF\lp 0\rp = 0$, and thus we may define $u\lp t,x\rp$ as the function given by:
 		\begin{align}
-			&\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \\
-			&= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 \\
-			&+\fR_3 \cdot \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp 
+			u\lp t,x\rp &= \exp \lp \ff\lp t\rp\rp \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \nonumber\\
+			&= \lb \exp\lp \fF\lp 0\rp\rp  + \int_0^s \ff'\lp s\rp\cdot \exp \lp \ff\lp s\rp\rp ds\rb \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\nonumber \\
+			&=\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_0^s \ff'\lp s\rp \cdot \exp\lp \ff\lp s\rp\rp \cdot \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp ds \nonumber\\
+			&= \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int^s_0 \ff'\lp s\rp\cdot u\lp s,\cX^{d,t,x}_{r,\Omega}\rp ds \nonumber \\
+			&=\fu^T\lp \cX^{d,t,x}\rp + \int^s_0 \fF \lp s,u\lp s,x + \cW^d_r\rp\rp
 		\end{align}
-	\end{proof}
-	Jensen's Inequality, the fact that $\fu^T$ does not depend on time, and the linearity of integrals gives us:
-	\begin{align}
-		&= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } ds + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp^2 \nonumber\\
-			&+\fR_3 \cdot \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \nonumber\\
-		&\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \frac{1}{2}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\cdot \lp \frac{1}{T-t}\int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds\rp^2 \\ &+ \fR_3\nonumber\\
-		&\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_t^T\fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds  + \int^T_t \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega }\rp \cdot \lp \alpha_d \circ  \cX^{d,t,x}_{r,\Omega }\rp^2 ds\\ &+ \fR_3\nonumber \\
-		&= \fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp + \int^T_t \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} + \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp\alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 ds + \fR_3\nonumber
-	\end{align}
-	Thus \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $f \curvearrowleft \fu^T$ tells us that:
-	\begin{align}
-		\E
-	\end{align}
+		Then \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $u \curvearrowleft u$, 
+%		Note that Taylor's theorem states that:
+%		\begin{align}
+%			\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp = 1 + \int^T_t \alpha_d \circ \cX ^{d,t,x}_{r,\Omega}ds + \frac{1}{2}\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega }\rp^2 ds + \fR_3
+%		\end{align}
+%		Where $\fR_3$ is the Lagrange form of the reamainder. Thus $\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot u\lp T,\cX^{d,t,x}_{r,\Omega}\rp$ is rendered as:
+%		\begin{align}
+%			&\exp\lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega } ds\rp \cdot \fu_d^T\lp\cX^{d,t,x}_{r,\Omega}\rp \\
+%			&= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 \\
+%			&+\fR_3 \cdot \fu_d^T\lp \cX^{d,t,s}_{r,\Omega}\rp 
+%		\end{align}
+%	\end{proof}
+%	Jensen's Inequality, the fact that $\fu^T$ does not depend on time, and the linearity of integrals gives us:
+%	\begin{align}
+%		&= \fu^T\lp\cX^{d,t,s}_{r,\Omega }\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\omega } ds + \frac{1}{2} \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega}ds\rp^2 \nonumber\\
+%			&+\fR_3 \cdot \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \nonumber\\
+%		&\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \fu^T\lp \cX^{d,t,s}_{r,\Omega}\rp \cdot \int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds + \frac{1}{2}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp\cdot \lp \frac{1}{T-t}\int^T_t \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds\rp^2 \\ &+ \fR_3\nonumber\\
+%		&\les \fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp + \int_t^T\fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} ds  + \int^T_t \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega }\rp \cdot \lp \alpha_d \circ  \cX^{d,t,x}_{r,\Omega }\rp^2 ds\\ &+ \fR_3\nonumber \\
+%		&= \fu^T\lp \cX^{d,t,x}_{r,\Omega} \rp + \int^T_t \fu^T \lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \alpha_d \circ \cX^{d,t,x}_{r,\Omega} + \frac{1}{2\lp T-t\rp}\fu^T\lp \cX^{d,t,x}_{r,\Omega}\rp \cdot \lp\alpha_d \circ \cX^{d,t,x}_{r,\Omega}\rp^2 ds + \fR_3\nonumber
+%	\end{align}
+%	Thus \cite[Lemma~2.3]{hutzenthaler_strong_2021} with $f \curvearrowleft \fu^T$ tells us that:
+%	\begin{align}
+%		\E
+%	\end{align}
 %	\begin{proof}
 %	Note that $\fu^T$ is deterministic, and $\cX^{d,t,x}_{r,\Omega}$ is a $d$-vector of random variables, where $\mu = \mymathbb{0}_d$, and $\Sigma = \mathbb{I}_d$. Whence we have that:
 %	\begin{align}
@@ -1038,7 +1055,7 @@ Let $t \in \lp 0,\infty\rp$ and $T \in \lp t,\infty\rp$. Let $\lp \Omega, \mathc
 %\end{align}
 %
 %
-%\end{proof}
+\end{proof}
 
 
 
diff --git a/Dissertation/main.pdf b/Dissertation/main.pdf
index 3b9e87b..eeafd34 100644
Binary files a/Dissertation/main.pdf and b/Dissertation/main.pdf differ
diff --git a/Dissertation/preamble.tex b/Dissertation/preamble.tex
index b3e0668..f5c2946 100644
--- a/Dissertation/preamble.tex
+++ b/Dissertation/preamble.tex
@@ -2,6 +2,7 @@
 \usepackage{setspace}
 \doublespacing
 \usepackage[toc,page]{appendix}
+\usepackage{mleftright}
 
 \usepackage{pdfpages}
 \usepackage[]{amsmath}