\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{t86\_quatern3 (TMYnGkDEnzdF8MD2FygARMgBrJDyXArMxyz)}
\maketitle
Let $ k4\_quatern3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k12\_quaterni : {\iota}$ be given.
Let $ k28\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k11\_quaterni : {\iota}$ be given.
Let $ k4\_quatern2 : {\iota}$ be given.
Let $ v1\_quaterni : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_quatern3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_quatern3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k27\_quaterni : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_quaterni : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k17\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_quatern2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ k31\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_xcmplx\_0 : {\iota}$ be given.
Let $ k18\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k19\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k20\_quaterni : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k6\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_quatern2 : {\iota}$ be given.
Let $ k10\_quaterni : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k7\_quatern2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_quaterni : {\iota}$ be given.
Let $ k3\_quatern3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k22\_quaterni : {\iota}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_quaterni : {\iota}$ be given.
Let $ k4\_quaterni : {\iota}$ be given.
Let $ k5\_arytm\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k5\_funct\_4 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
{{{k4\_quatern3}~{k11\_quaterni}}={{k28\_quaterni}~{k11\_quaterni}}}\label{hyp:ehidfdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{{k4\_quatern3}~{k4\_quatern2}}={{k28\_quaterni}~{k4\_quatern2}}}\label{hyp:ehidedpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k1\_quatern3~X0=k2\_quatern3~(k28\_quaterni~\\
X0))\end{array}
\label{hyp:ehidbdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}((m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(k27\_quaterni~\\
X0~k12\_quaterni=k6\_quaterni~k6\_numbers~k6\_numbers~k6\_numbers~\\
(k17\_quaterni~X0)))\end{array}
\label{hyp:ehhdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{{k8\_quatern2}~{k12\_quaterni}}={{k28\_quaterni}~{k12\_quaterni}}}\label{hyp:ehhdhdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xboole\_0}~{X0}})}{\unicdbdjdcd}{({{X0}={k1\_xboole\_0}})}}}\label{hyp:ehgdpfcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k31\_quaterni~(k28\_quaterni~X0)=\\
k28\_quaterni~(k31\_quaterni~X0))\end{array}
\label{hyp:ehfdfdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{{k31\_quaterni}~{k12\_quaterni}}={{k28\_quaterni}~{k12\_quaterni}}}\label{hyp:ehfdddpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{{k31\_quaterni}~{k11\_quaterni}}={{k28\_quaterni}~{k11\_quaterni}}}\label{hyp:ehfdcdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{{k31\_quaterni}~{k1\_xcmplx\_0}}={{k28\_quaterni}~{k1\_xcmplx\_0}}}\label{hyp:ehfdbdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k17\_quaterni~(k31\_quaterni~k12\_quaterni)=k6\_numbers){\unicdcdcdhd}((k18\_quaterni~\\
(k31\_quaterni~k12\_quaterni)=k6\_numbers){\unicdcdcdhd}((k19\_quaterni~(k31\_quaterni~\\
k12\_quaterni)=k6\_numbers){\unicdcdcdhd}(k20\_quaterni~(k31\_quaterni~k12\_quaterni)=\\
k1\_real\_1~np\_\_1)))\end{array}
\label{hyp:ehfdadpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k6\_xcmplx\_0}~{X0}}~{k6\_numbers}}={X0}})}}}\label{hyp:ehedpfbgchjgehigng}
\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}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}((k17\_quaterni~(k28\_quaterni~X0)=\\
k1\_real\_1~(k17\_quaterni~X0)){\unicdcdcdhd}((k18\_quaterni~(k28\_quaterni~X0)=\\
k1\_real\_1~(k18\_quaterni~X0)){\unicdcdcdhd}((k19\_quaterni~(k28\_quaterni~X0)=\\
k1\_real\_1~(k19\_quaterni~X0)){\unicdcdcdhd}(k20\_quaterni~(k28\_quaterni~X0)=\\
k1\_real\_1~(k20\_quaterni~X0)))))\end{array}
\label{hyp:ehedbdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k8\_quatern2~(k28\_quaterni~X0)=\\
k28\_quaterni~(k8\_quatern2~X0))\end{array}
\label{hyp:ehddedpfbhfhbgehfgchogcd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(X0=k6\_quaterni~(k17\_quaterni~X0)~\\
(k18\_quaterni~X0)~(k19\_quaterni~X0)~(k20\_quaterni~X0))\end{array}
\label{hyp:ehcdedpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_quaterni~X1){\unicdbdjdcd}(k27\_quaterni~\\
(k28\_quaterni~X0)~(k28\_quaterni~X1)=k27\_quaterni~X0~X1))\end{array}
\label{hyp:ehcdcdpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_quaterni~X1){\unicdbdjdcd}(k27\_quaterni~\\
X0~(k28\_quaterni~X1)=k28\_quaterni~(k27\_quaterni~X0~X1)))\end{array}
\label{hyp:ehcdbdpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_quaterni~X1){\unicdbdjdcd}(k27\_quaterni~\\
(k28\_quaterni~X0)~X1=k28\_quaterni~(k27\_quaterni~X0~X1)))\end{array}
\label{hyp:ehcdadpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k2\_xcmplx\_0}~{X0}}~{k6\_numbers}}={X0}})}}}\label{hyp:ehbdpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k28\_quaterni~X0=k27\_quaterni~(\\
k28\_quaterni~k2\_quatern2)~X0)\end{array}
\label{hyp:ehbdjdpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
{{{{k10\_quaterni}~{k12\_quaterni}}~{k12\_quaterni}}={{k1\_real\_1}~{np\_\_1}}}\label{hyp:ehbdfdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_quaterni}~{X0}})}{\unicdbdjdcd}{({{{{k27\_quaterni}~{k2\_quatern2}}~{X0}}={X0}})}}}\label{hyp:ehbdfdpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}((v2\_xxreal\_0~np\_\_1){\unicdcdcdhd}(m2\_subset\_1~np\_\_1~k1\_numbers~k5\_numbers)){\unicdcdcdhd}\\
((m1\_subset\_1~np\_\_1~k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_1~k1\_numbers))\end{array}
\label{hyp:dhahdgbdpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_quaterni}~{X0}})}{\unicdbdjdcd}{({{{k8\_quatern2}~{X0}}={{k7\_quatern2}~{X0}}})}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgidpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_numbers}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_quaterni){\unicdbdjdcd}(k4\_quatern3~X0=k3\_quatern3~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgedpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{k4\_quatern2}={k1\_xcmplx\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgedpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_quaterni){\unicdbdjdcd}(k2\_quatern3~X0=k1\_quatern3~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{k2\_quatern2}={k22\_quaterni}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_quaterni~X0){\unicdcdcdhd}(v1\_quaterni~X1)){\unicdbdjdcd}(\\
k27\_quaterni~X0~X1=k10\_quaterni~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdhdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{{k1\_real\_1}~{X0}}={{k4\_xcmplx\_0}~{X0}}})}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k12\_quaterni}={k5\_quaterni}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdcdpfbhfhbgehfgchogjg}
\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}((m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}((\\
X0=k17\_quaterni~X0){\unicdcdcdhd}((k18\_quaterni~X0=k6\_numbers){\unicdcdcdhd}((k19\_quaterni~\\
X0=k6\_numbers){\unicdcdcdhd}(k20\_quaterni~X0=k6\_numbers)))))\end{array}
\label{hyp:mgcdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k1\_numbers){\unicdbdjdcd}(k6\_quaterni~X0~X1~k6\_numbers~k6\_numbers=k5\_arytm\_0~\\
X0~X1))\end{array}
\label{hyp:mgbdfdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_quaterni}~{X0}})}{\unicdbdjdcd}{({{{k28\_quaterni}~{({{k28\_quaterni}~{X0}})}}={X0}})}}}\label{hyp:jgogghpgmgfhehjgghfgogfgdhdhpflgcdidpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{{k1\_real\_1}~{({{k1\_real\_1}~{X0}})}}={X0}})}}}\label{hyp:jgogghpgmgfhehjgghfgogfgdhdhpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_quaterni~X0){\unicdcdcdhd}(v1\_quaterni~X1)){\unicdbdjdcd}(\\
v1\_quaterni~(k10\_quaterni~X0~X1))\end{array}
\label{hyp:ggdghdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_quaterni}~{k5\_quaterni}}\label{hyp:ggdgfdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_quaterni}~{k4\_quaterni}}\label{hyp:ggdgedpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((v1\_xcmplx\_0~(k4\_xcmplx\_0~X0)){\unicdcdcdhd}\\
(v1\_xreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgddpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_quaterni}~{k1\_xcmplx\_0}}\label{hyp:ggdgddpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xboole\_0}~{k1\_xboole\_0}}\label{hyp:ggdgbdpfihcgpgpgmgfgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v3\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X0)){\unicdbdjdcd}((v1\_xcmplx\_0~\\
(k4\_xcmplx\_0~X0)){\unicdcdcdhd}(\preoh v2\_xxreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgbdgdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X0)){\unicdbdjdcd}((v1\_xcmplx\_0~\\
(k4\_xcmplx\_0~X0)){\unicdcdcdhd}(\preoh v3\_xxreal\_0~(k4\_xcmplx\_0~X0)))\end{array}
\label{hyp:ggdgbdfdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_quaterni}~{X0}})}{\unicdbdjdcd}{({{v1\_quaterni}~{({{k7\_quatern2}~{X0}})}})}}}\label{hyp:egehpflghdpfbhfhbgehfgchogcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{({{k4\_xcmplx\_0}~{X0}})}})}}}\label{hyp:egehpflgedpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(m1\_subset\_1~(k31\_quaterni~X0)~k1\_quaterni)\end{array}
\label{hyp:egehpflgddbdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(m1\_subset\_1~(k28\_quaterni~X0)~k1\_quaterni)\end{array}
\label{hyp:egehpflgcdidpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_quaterni~X0){\unicdcdcdhd}(v1\_quaterni~X1)){\unicdbdjdcd}(\\
m1\_subset\_1~(k27\_quaterni~X0~X1)~k1\_quaterni)\end{array}
\label{hyp:egehpflgcdhdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_quaterni}~{k22\_quaterni}}\label{hyp:egehpflgcdcdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(m1\_subset\_1~(k20\_quaterni~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgcdadpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(m1\_subset\_1~(k1\_real\_1~\\
X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(m1\_subset\_1~(k19\_quaterni~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdjdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(m1\_subset\_1~(k18\_quaterni~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdidpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(m1\_subset\_1~(k17\_quaterni~X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdhdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{{m1\_subset\_1}~{k12\_quaterni}}~{k1\_quaterni}}\label{hyp:egehpflgbdcdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}(k6\_xcmplx\_0~\\
X0~X1=k2\_xcmplx\_0~X0~(k4\_xcmplx\_0~X1)))\end{array}
\label{hyp:egidpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~k1\_numbers){\unicdbdjdcd}(((X1=k6\_numbers){\unicdbdjdcd}(k5\_arytm\_0~X0~X1=X0)){\unicdcdcdhd}((\\
X1{\unicdcdgdad}k6\_numbers){\unicdbdjdcd}(k5\_arytm\_0~X0~X1=k5\_funct\_4~k1\_numbers~k6\_numbers~\\
np\_\_1~X0~X1))))\end{array}
\label{hyp:egfdpfbgchjhehngpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_quaterni~X0){\unicdbdjdcd}(k3\_quatern3~X0=k27\_quaterni~(k27\_quaterni~\\
X0~X0)~X0)\end{array}
\label{hyp:egcdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_quaterni}~{X0}})}{\unicdbdjdcd}{({{{k1\_quatern3}~{X0}}={{{k27\_quaterni}~{X0}}~{X0}}})}}}\label{hyp:egbdpfbhfhbgehfgchogdd}
\end{equation}
Assume the following.
\begin{equation}
{{k22\_quaterni}={np\_\_1}}\label{hyp:egbdidpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
{{k12\_quaterni}={{{{{k6\_quaterni}~{k6\_numbers}}~{k6\_numbers}}~{k6\_numbers}}~{np\_\_1}}}\label{hyp:egbdcdpfbhfhbgehfgchogjg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
k2\_xcmplx\_0~X0~X1=k2\_xcmplx\_0~X1~X0)\end{array}
\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgcdpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}((\preoh v2\_xxreal\_0~X0){\unicdcdcdhd}(\preoh v3\_xxreal\_0~\\
X0))){\unicdbdjdcd}((v1\_xboole\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X0))\end{array}
\label{hyp:dgdgidpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xxreal\_0}~{X0}})}}}\label{hyp:dgdgedpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v2\_xxreal\_0~X0)){\unicdbdjdcd}((\preoh v1\_xboole\_0~\\
X0){\unicdcdcdhd}((v1\_xxreal\_0~X0){\unicdcdcdhd}(\preoh v3\_xxreal\_0~X0)))\end{array}
\label{hyp:dgdgddpfihihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_numbers}})}{\unicdbdjdcd}{({{v1\_xreal\_0}~{X0}})}}}\label{hyp:dgdgbdpfihchfgbgmgpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k1\_quaterni}})}{\unicdbdjdcd}{({{v1\_quaterni}~{X0}})}}}\label{hyp:dgdgbdpfbhfhbgehfgchogjg}
\end{equation}
\begin{theorem}\label{thm:ehidgdpfbhfhbgehfgchogdd}
${{{k4\_quatern3}~{k12\_quaterni}}={{k28\_quaterni}~{k12\_quaterni}}}$.
\end{theorem}
\end{document}
