\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{l34\_sin\_cos8 (TMbYvYNTd4EeUTR5zLcWj5V8tvjBWGTjzPR)}
\maketitle
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k7\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k9\_sin\_cos2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ np\_\_1 : {\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 $ k1\_numbers : {\iota}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ np\_\_0 : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xcmplx\_0~X2){\unicdbdjdcd}((X0{\unicdcdgdad}k6\_numbers){\unicdbdjdcd}(k7\_xcmplx\_0~X1~X2=k7\_xcmplx\_0~\\
(k3\_xcmplx\_0~X1~X0)~(k3\_xcmplx\_0~X2~X0)))))\end{array}
\label{hyp:ehjdbdpfihdgngahmgihpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}((\\
X0{\unicdcdgdad}k6\_numbers){\unicdbdjdcd}(k3\_xcmplx\_0~(k7\_xcmplx\_0~X1~X0)~X0=X1)))\end{array}
\label{hyp:ehidhdpfihdgngahmgihpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xcmplx\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xcmplx\_0~X2){\unicdbdjdcd}({\unicdcdadad} X3 .  (v1\_xcmplx\_0~X3){\unicdbdjdcd}(k3\_xcmplx\_0~\\
(k7\_xcmplx\_0~X0~X1)~(k7\_xcmplx\_0~X2~X3)=k7\_xcmplx\_0~(k3\_xcmplx\_0~\\
X0~X2)~(k3\_xcmplx\_0~X1~X3)))))\end{array}
\label{hyp:ehhdgdpfihdgngahmgihpfbd}
\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}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{{k3\_xcmplx\_0}~{np\_\_1}}~{X0}}={X0}})}}}\label{hyp:ehddpfbgchjgehigng}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((k9\_sin\_cos2~X0=k7\_xcmplx\_0~(k3\_sin\_cos2~\\
X0)~(k6\_sin\_cos2~X0)){\unicdcdcdhd}(k9\_sin\_cos2~k6\_numbers=k6\_numbers))\end{array}
\label{hyp:ehbdpfdhjgogpfdgpgdhid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}((v1\_xcmplx\_0~\\
X1){\unicdcdcdhd}(v1\_xcmplx\_0~X2))){\unicdbdjdcd}(k3\_xcmplx\_0~(k3\_xcmplx\_0~X0~X1)~X2=k3\_xcmplx\_0~\\
X0~(k3\_xcmplx\_0~X1~X2))\end{array}
\label{hyp:dhahdghdpfbgchjgehigng}
\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}
{{v1\_xboole\_0}~{np\_\_0}}\label{hyp:dhahdgadpfcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
{{{{k7\_xcmplx\_0}~{np\_\_1}}~{np\_\_1}}={np\_\_1}}\label{hyp:chbhcffgbgmgeejgghpfpflghdpfihdgngahmgihpfadpfpfchbdpfchbdpfchbd}
\end{equation}
Assume the following.
\begin{equation}
{\preoh{{{{r1\_xxreal\_0}~{np\_\_1}}~{np\_\_0}}}}\label{hyp:chbhmefgdhdhpechfebhfhbgmgpfpfchbdpfihihchfgbgmgpfadpfpfchbdpfchad}
\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 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((r1\_xxreal\_0~np\_\_1~(k6\_sin\_cos2~\\
X0)){\unicdcdcdhd}((k6\_sin\_cos2~k6\_numbers=np\_\_1){\unicdcdcdhd}(k3\_sin\_cos2~k6\_numbers=\\
k6\_numbers)))\end{array}
\label{hyp:mgbdpfdhjgogpfdgpgdhid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xreal\_0~X1)){\unicdbdjdcd}(v1\_xreal\_0~\\
(k3\_xcmplx\_0~X0~X1))\end{array}
\label{hyp:ggdggdpfihchfgbgmgpfad}
\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}~{({{k6\_sin\_cos2}~{X0}})}}~{k1\_numbers}}}\label{hyp:egehpflggdpfdhjgogpfdgpgdhcd}
\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}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xcmplx\_0~X0){\unicdcdcdhd}(v1\_xcmplx\_0~X1)){\unicdbdjdcd}(\\
k3\_xcmplx\_0~X0~X1=k3\_xcmplx\_0~X1~X0)\end{array}
\label{hyp:dgpgngngfhehbgehjgghjgehjhpflgddpfihdgngahmgihpfad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xreal\_0}~{X0}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgddpfihchfgbgmgpfad}
\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}
\begin{theorem}\label{thm:mgddedpfdhjgogpfdgpgdhid}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xreal\_0~X1){\unicdbdjdcd}((k7\_xcmplx\_0~\\
(k3\_xcmplx\_0~(k3\_sin\_cos2~X0)~(k6\_sin\_cos2~X1))~(k3\_xcmplx\_0~\\
(k6\_sin\_cos2~X0)~(k6\_sin\_cos2~X1))=k9\_sin\_cos2~X0){\unicdcdcdhd}((k3\_xcmplx\_0~\\
(k3\_sin\_cos2~X0)~(k6\_sin\_cos2~X1)=k3\_xcmplx\_0~(k9\_sin\_cos2~X0)~\\
(k3\_xcmplx\_0~(k6\_sin\_cos2~X0)~(k6\_sin\_cos2~X1))){\unicdcdcdhd}((k3\_sin\_cos2~\\
X0=k3\_xcmplx\_0~(k9\_sin\_cos2~X0)~(k6\_sin\_cos2~X0)){\unicdcdcdhd}(k3\_xcmplx\_0~\\
(k3\_sin\_cos2~X0)~(k3\_sin\_cos2~X1)=k3\_xcmplx\_0~(k3\_xcmplx\_0~(\\
k9\_sin\_cos2~X0)~(k9\_sin\_cos2~X1))~(k3\_xcmplx\_0~(k6\_sin\_cos2~X0)~\\
(k6\_sin\_cos2~X1)))))))\end{array}$$
\end{theorem}
\end{document}
