\documentclass{article}
\newtheorem{axiom}{Axiom}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
\newcommand\unicdcdcdbg{\cup}
\newcommand\unicdcdbdgd{\setminus}
\newcommand\unicdcdcdjd{\cap}
\newcommand\unicdcdadad{\forall}
\newcommand\unicdbdjdcd{\Rightarrow}
\newcommand\unicdcdadjd{\notin}
\newcommand\unicdbdjded{\Leftrightarrow}
\newcommand\unicdcdcdid{\lor}
\newcommand\unicdcdcdhd{\land}
\newcommand\uniddcgcg{\lambda}
\newcommand\unicdcdaddd{\exists}
\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
\newcommand\mname[1]{{\mathsf{#1}}}
\begin{document}
\title{t15\_gate\_1 (TMNsxux2jfyKWDoFkUmNEMifr2iWYxRkdo6)}
\maketitle
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k9\_gate\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}(\preoh v1\_xboole\_0~\\
(k9\_gate\_1~X1~X2~X0))\end{array}
\label{hyp:ggdgcdedpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}(\preoh v1\_xboole\_0~\\
(k9\_gate\_1~X1~X0~X2))\end{array}
\label{hyp:ggdgcdddpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~X0){\unicdbdjdcd}(\preoh v1\_xboole\_0~\\
(k9\_gate\_1~X0~X1~X2))\end{array}
\label{hyp:ggdgcdcdpfhgbgehfgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((v1\_xboole\_0~X0){\unicdcdcdhd}((v1\_xboole\_0~\\
X1){\unicdcdcdhd}(v1\_xboole\_0~X2))){\unicdbdjdcd}(v1\_xboole\_0~(k9\_gate\_1~X0~X1~X2))\end{array}
\label{hyp:ggdgcdbdpfhgbgehfgpfbd}
\end{equation}
\begin{theorem}\label{thm:ehbdfdpfhgbgehfgpfbd}
$$\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (\preoh (\preoh v1\_xboole\_0~(k9\_gate\_1~\\
X0~X1~X2)){\unicdcdcdhd}((v1\_xboole\_0~X0){\unicdcdcdhd}((v1\_xboole\_0~X1){\unicdcdcdhd}(v1\_xboole\_0~\\
X2)))){\unicdcdcdhd}(\preoh (\preoh (v1\_xboole\_0~X0){\unicdcdcdhd}((v1\_xboole\_0~X1){\unicdcdcdhd}(v1\_xboole\_0~\\
X2))){\unicdcdcdhd}(v1\_xboole\_0~(k9\_gate\_1~X0~X1~X2)))\end{array}$$
\end{theorem}
\end{document}
