\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{t52\_quaterni (TMZDgkQJNxNcNUBb2n6bZkr2v5TJeUWfMbz)}
\maketitle
Let $ k31\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k11\_quaterni : {\iota}$ be given.
Let $ k28\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k17\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_xcmplx\_0 : {\iota}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k18\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k19\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k20\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_quaterni : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k12\_quaterni : {\iota}$ be given.
Let $ k30\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_quaterni : {\iota}$ be given.
Let $ k26\_quaterni : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k23\_quaterni : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k25\_quaterni : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}(k17\_quaterni~(k31\_quaterni~k1\_xcmplx\_0)=k6\_numbers){\unicdcdcdhd}((k18\_quaterni~\\
(k31\_quaterni~k1\_xcmplx\_0)=k1\_real\_1~np\_\_1){\unicdcdcdhd}((k19\_quaterni~\\
(k31\_quaterni~k1\_xcmplx\_0)=k6\_numbers){\unicdcdcdhd}(k20\_quaterni~(k31\_quaterni~\\
k1\_xcmplx\_0)=k6\_numbers)))\end{array}
\label{hyp:ehedidpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}((k17\_quaterni~(k31\_quaterni~X0)=\\
k17\_quaterni~X0){\unicdcdcdhd}((k18\_quaterni~(k31\_quaterni~X0)=k1\_real\_1~\\
(k18\_quaterni~X0)){\unicdcdcdhd}((k19\_quaterni~(k31\_quaterni~X0)=k1\_real\_1~\\
(k19\_quaterni~X0)){\unicdcdcdhd}(k20\_quaterni~(k31\_quaterni~X0)=k1\_real\_1~\\
(k20\_quaterni~X0)))))\end{array}
\label{hyp:ehededpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k17\_quaterni~k11\_quaterni=k6\_numbers){\unicdcdcdhd}((k18\_quaterni~k11\_quaterni=\\
k6\_numbers){\unicdcdcdhd}((k19\_quaterni~k11\_quaterni=np\_\_1){\unicdcdcdhd}((k20\_quaterni~\\
k11\_quaterni=k6\_numbers){\unicdcdcdhd}((k17\_quaterni~k12\_quaterni=k6\_numbers){\unicdcdcdhd}\\
((k18\_quaterni~k12\_quaterni=k6\_numbers){\unicdcdcdhd}((k19\_quaterni~k12\_quaterni=\\
k6\_numbers){\unicdcdcdhd}(k20\_quaterni~k12\_quaterni=np\_\_1)))))))\end{array}
\label{hyp:ehddbdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k17\_quaterni~k1\_xcmplx\_0=k6\_numbers){\unicdcdcdhd}((k18\_quaterni~k1\_xcmplx\_0=\\
np\_\_1){\unicdcdcdhd}((k19\_quaterni~k1\_xcmplx\_0=k6\_numbers){\unicdcdcdhd}(k20\_quaterni~\\
k1\_xcmplx\_0=k6\_numbers)))\end{array}
\label{hyp:ehddadpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_quaterni}~{X0}})}{\unicdbdjdcd}{({{{k31\_quaterni}~{X0}}={{k30\_quaterni}~{X0}}})}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgddbdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{k11\_quaterni}={k4\_quaterni}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdbdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k28\_quaterni~X0=k26\_quaterni~(\\
k26\_quaterni~(k23\_quaterni~(k1\_real\_1~(k17\_quaterni~X0))~(k25\_quaterni~\\
(k1\_real\_1~(k18\_quaterni~X0))~k1\_xcmplx\_0))~(k25\_quaterni~(k1\_real\_1~\\
(k19\_quaterni~X0))~k11\_quaterni))~(k25\_quaterni~(k1\_real\_1~(\\
k20\_quaterni~X0))~k12\_quaterni))\end{array}
\label{hyp:mgidgdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_quaterni}~{k4\_quaterni}}\label{hyp:ggdgedpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_quaterni}~{k1\_xcmplx\_0}}\label{hyp:ggdgddpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k30\_quaterni~X0=k26\_quaterni~(\\
k26\_quaterni~(k23\_quaterni~(k17\_quaterni~X0)~(k25\_quaterni~(\\
k1\_real\_1~(k18\_quaterni~X0))~k1\_xcmplx\_0))~(k25\_quaterni~(k1\_real\_1~\\
(k19\_quaterni~X0))~k11\_quaterni))~(k25\_quaterni~(k1\_real\_1~(\\
k20\_quaterni~X0))~k12\_quaterni))\end{array}
\label{hyp:egcdcdpfbhfhbgehfgchogjg}
\end{equation}
\begin{theorem}\label{thm:ehfdcdpfbhfhbgehfgchogjg}
${{{k31\_quaterni}~{k11\_quaterni}}={{k28\_quaterni}~{k11\_quaterni}}}$.
\end{theorem}
\end{document}
