\documentclass{article}
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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{definition}{Definition}
\newtheorem{example}{Example}
\newtheorem{conjecture}{Conjecture}
\newcommand\unicdcdgdad{\not=}
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\newcommand\unicdbdjded{\Leftrightarrow}
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\newcommand\unicdcdidid{\not\subseteq}
\newcommand\preoh[1]{\lnot #1}
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\begin{document}
\title{l9\_sin\_cos8 (TMMdiJzjr1cR8bLhmWyAYp1X84A6NYQoYME)}
\maketitle
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k9\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_sin\_cos5 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_sin\_cos5 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_sin\_cos5 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_sin\_cos2 : {\iota}$ be given.
Let $ k7\_binop\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_sin\_cos2 : {\iota}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k7\_sin\_cos2 : {\iota}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k7\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v2\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k8\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k10\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((k1\_seq\_1~k4\_sin\_cos2~(k7\_binop\_2~\\
X0)=k1\_seq\_1~k4\_sin\_cos2~X0){\unicdcdcdhd}((k1\_seq\_1~k1\_sin\_cos2~(k7\_binop\_2~\\
X0)=k1\_real\_1~(k1\_seq\_1~k1\_sin\_cos2~X0)){\unicdcdcdhd}(k1\_seq\_1~k7\_sin\_cos2~\\
(k7\_binop\_2~X0)=k1\_real\_1~(k1\_seq\_1~k7\_sin\_cos2~X0))))\end{array}
\label{hyp:ehbdjdpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}(k7\_xcmplx\_0~\\
X0~(k4\_xcmplx\_0~X1)=k4\_xcmplx\_0~(k7\_xcmplx\_0~X0~X1)))\end{array}
\label{hyp:ehbdididpfihdgngahmgihpfbd}
\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 .  {{{k9\_sin\_cos2}~{X0}}={{k8\_sin\_cos2}~{X0}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgjdpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{{k7\_binop\_2}~{X0}}={{k4\_xcmplx\_0}~{X0}}})}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflghdpfcgjgogpgahpfcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k6\_sin\_cos2}~{X0}}={{k5\_sin\_cos2}~{X0}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k3\_sin\_cos2}~{X0}}={{k2\_sin\_cos2}~{X0}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfdhjgogpfdgpgdhcd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k1\_numbers){\unicdcdcdhd}(v1\_xreal\_0~\\
X1)){\unicdbdjdcd}(k10\_real\_1~X0~X1=k7\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdadpfchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(k1\_sin\_cos5~(k4\_xcmplx\_0~X0)=k1\_real\_1~\\
(k1\_sin\_cos5~X0))\end{array}
\label{hyp:mgedddpfdhjgogpfdgpgdhfd}
\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}
{{\unicdcdadad} X0 .  {{{m1\_subset\_1}~{({{k9\_sin\_cos2}~{X0}})}}~{k1\_numbers}}}\label{hyp:egehpflgjdpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{m1\_subset\_1}~{({{k3\_sin\_cos2}~{X0}})}}~{k1\_numbers}}}\label{hyp:egehpflgddpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{{m1\_subset\_1}~{({{k1\_sin\_cos5}~{X0}})}}~{k1\_numbers}})}}}\label{hyp:egehpflgbdpfdhjgogpfdgpgdhfd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k8\_sin\_cos2}~{X0}}={{{k1\_seq\_1}~{k7\_sin\_cos2}}~{X0}}}}\label{hyp:eggdpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k5\_sin\_cos2}~{X0}}={{{k1\_seq\_1}~{k4\_sin\_cos2}}~{X0}}}}\label{hyp:egedpfdhjgogpfdgpgdhcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(k3\_sin\_cos5~X0=k10\_real\_1~np\_\_1~\\
(k3\_sin\_cos2~X0))\end{array}
\label{hyp:egddpfdhjgogpfdgpgdhfd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}(k2\_sin\_cos5~X0=k10\_real\_1~np\_\_1~\\
(k6\_sin\_cos2~X0))\end{array}
\label{hyp:egcdpfdhjgogpfdgpgdhfd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k2\_sin\_cos2}~{X0}}={{{k1\_seq\_1}~{k1\_sin\_cos2}}~{X0}}}}\label{hyp:egcdpfdhjgogpfdgpgdhcd}
\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\_numbers}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgbdpfihdgngahmgihpfad}
\end{equation}
\begin{theorem}\label{thm:mgjdpfdhjgogpfdgpgdhid}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((k3\_sin\_cos2~(k4\_xcmplx\_0~X0)=k4\_xcmplx\_0~\\
(k3\_sin\_cos2~X0)){\unicdcdcdhd}((k6\_sin\_cos2~(k4\_xcmplx\_0~X0)=k6\_sin\_cos2~\\
X0){\unicdcdcdhd}((k9\_sin\_cos2~(k4\_xcmplx\_0~X0)=k4\_xcmplx\_0~(k9\_sin\_cos2~\\
X0)){\unicdcdcdhd}((k1\_sin\_cos5~(k4\_xcmplx\_0~X0)=k4\_xcmplx\_0~(k1\_sin\_cos5~\\
X0)){\unicdcdcdhd}((k2\_sin\_cos5~(k4\_xcmplx\_0~X0)=k2\_sin\_cos5~X0){\unicdcdcdhd}(k3\_sin\_cos5~\\
(k4\_xcmplx\_0~X0)=k4\_xcmplx\_0~(k3\_sin\_cos5~X0)))))))\end{array}$$
\end{theorem}
\end{document}
