\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{t10\_integra8 (TMYZKmtooo89kLG6p5EwZMycMn9RJTfHx7f)}
\maketitle
Let $ k3\_sin\_cos6 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k10\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_square\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ k32\_sin\_cos : {\iota}$ be given.
Let $ np\_\_4 : {\iota}$ be given.
Let $ k18\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_real\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k8\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k17\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k21\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k20\_sin\_cos : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k9\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_real\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_seq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k19\_sin\_cos : {\iota}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k16\_sin\_cos : {\iota}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_xboole\_0 : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k4\_xxreal\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_xxreal\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k5\_sin\_cos9 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_sin\_cos9 : {\iota}$ be given.
Let $ k1\_fdiff\_9 : {\iota}$ be given.
Let $ np\_\_3 : {\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 $ k2\_sin\_cos6 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_rcomp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_zfmisc\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_sin\_cos6 : {\iota}$ be given.
Let $ v2\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}k18\_sin\_cos~(k1\_real\_1~(k10\_real\_1~k32\_sin\_cos~np\_\_4))=k1\_real\_1~\\
(k10\_real\_1~(k7\_square\_1~np\_\_2)~np\_\_2)\end{array}
\label{hyp:ehidpfjgogehfghgchbgid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}k18\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~np\_\_4)=k10\_real\_1~(k7\_square\_1~\\
np\_\_2)~np\_\_2\end{array}
\label{hyp:ehhdpfjgogehfghgchbgid}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((k18\_sin\_cos~(k3\_real\_1~X0~(k8\_real\_1~\\
np\_\_2~k32\_sin\_cos))=k17\_sin\_cos~X0){\unicdcdcdhd}((k21\_sin\_cos~(k3\_real\_1~\\
X0~(k8\_real\_1~np\_\_2~k32\_sin\_cos))=k20\_sin\_cos~X0){\unicdcdcdhd}((k18\_sin\_cos~\\
(k9\_real\_1~(k10\_real\_1~k32\_sin\_cos~np\_\_2)~X0)=k20\_sin\_cos~X0){\unicdcdcdhd}\\
((k21\_sin\_cos~(k9\_real\_1~(k10\_real\_1~k32\_sin\_cos~np\_\_2)~X0)=\\
k17\_sin\_cos~X0){\unicdcdcdhd}((k18\_sin\_cos~(k7\_real\_1~(k10\_real\_1~k32\_sin\_cos~\\
np\_\_2)~X0)=k20\_sin\_cos~X0){\unicdcdcdhd}((k21\_sin\_cos~(k7\_real\_1~(k10\_real\_1~\\
k32\_sin\_cos~np\_\_2)~X0)=k4\_xcmplx\_0~(k17\_sin\_cos~X0)){\unicdcdcdhd}((k18\_sin\_cos~\\
(k7\_real\_1~k32\_sin\_cos~X0)=k4\_xcmplx\_0~(k17\_sin\_cos~X0)){\unicdcdcdhd}(k21\_sin\_cos~\\
(k7\_real\_1~k32\_sin\_cos~X0)=k4\_xcmplx\_0~(k20\_sin\_cos~X0)))))))))\end{array}
\label{hyp:ehhdjdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k1\_seq\_1~k19\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~np\_\_2)=k6\_numbers){\unicdcdcdhd}\\
((k1\_seq\_1~k16\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~np\_\_2)=np\_\_1){\unicdcdcdhd}\\
((k1\_seq\_1~k19\_sin\_cos~k32\_sin\_cos=k1\_real\_1~np\_\_1){\unicdcdcdhd}((k1\_seq\_1~\\
k16\_sin\_cos~k32\_sin\_cos=k6\_numbers){\unicdcdcdhd}((k1\_seq\_1~k19\_sin\_cos~\\
(k7\_real\_1~k32\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~np\_\_2))=k6\_numbers){\unicdcdcdhd}\\
((k1\_seq\_1~k16\_sin\_cos~(k7\_real\_1~k32\_sin\_cos~(k10\_real\_1~k32\_sin\_cos~\\
np\_\_2))=k1\_real\_1~np\_\_1){\unicdcdcdhd}((k1\_seq\_1~k19\_sin\_cos~(k8\_real\_1~\\
np\_\_2~k32\_sin\_cos)=np\_\_1){\unicdcdcdhd}(k1\_seq\_1~k16\_sin\_cos~(k8\_real\_1~\\
np\_\_2~k32\_sin\_cos)=k6\_numbers)))))))\end{array}
\label{hyp:ehhdgdpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
{{{k3\_sin\_cos6}~{np\_\_1}}={{{k10\_real\_1}~{k32\_sin\_cos}}~{np\_\_2}}}\label{hyp:ehhdcdpfdhjgogpfdgpgdhgd}
\end{equation}
Assume the following.
\begin{equation}
{{{k3\_sin\_cos6}~{({{k1\_real\_1}~{np\_\_1}})}}={{k1\_real\_1}~{({{{k10\_real\_1}~{k32\_sin\_cos}}~{np\_\_2}})}}}\label{hyp:ehhdadpfdhjgogpfdgpgdhgd}
\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\_xreal\_0~X0){\unicdbdjdcd}(((r1\_xxreal\_0~(k1\_real\_1~(k10\_real\_1~\\
k32\_sin\_cos~np\_\_2))~X0){\unicdcdcdhd}(r1\_xxreal\_0~X0~(k10\_real\_1~k32\_sin\_cos~\\
np\_\_2))){\unicdbdjdcd}(k3\_sin\_cos6~(k17\_sin\_cos~X0)=X0))\end{array}
\label{hyp:ehgdjdpfdhjgogpfdgpgdhgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}((X0 \in k4\_xxreal\_1~X1~X2){\unicdbdjded}((\preoh r1\_xxreal\_0~X0~X1){\unicdcdcdhd}\\
(\preoh r1\_xxreal\_0~X2~X0)))))\end{array}
\label{hyp:ehedpfihihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xxreal\_0~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (v1\_xxreal\_0~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(v1\_xxreal\_0~X2){\unicdbdjdcd}((X0 \in k2\_xxreal\_1~X1~X2){\unicdbdjded}((r1\_xxreal\_0~X1~X0){\unicdcdcdhd}\\
(\preoh r1\_xxreal\_0~X2~X0)))))\end{array}
\label{hyp:ehddpfihihchfgbgmgpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k5\_sin\_cos9~np\_\_1=k10\_real\_1~k32\_sin\_cos~np\_\_4){\unicdcdcdhd}(k1\_seq\_1~\\
k1\_sin\_cos9~np\_\_1=k10\_real\_1~k32\_sin\_cos~np\_\_4)\end{array}
\label{hyp:ehddjdpfdhjgogpfdgpgdhjd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k5\_sin\_cos9~(k1\_real\_1~np\_\_1)=k1\_real\_1~(k10\_real\_1~k32\_sin\_cos~\\
np\_\_4)){\unicdcdcdhd}(k1\_seq\_1~k1\_sin\_cos9~(k1\_real\_1~np\_\_1)=k1\_real\_1~(\\
k10\_real\_1~k32\_sin\_cos~np\_\_4))\end{array}
\label{hyp:ehddhdpfdhjgogpfdgpgdhjd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k1\_seq\_1~k1\_fdiff\_9~k6\_numbers=np\_\_1){\unicdcdcdhd}((k1\_seq\_1~k1\_fdiff\_9~\\
(k10\_real\_1~k32\_sin\_cos~np\_\_4)=k7\_square\_1~np\_\_2){\unicdcdcdhd}((k1\_seq\_1~\\
k1\_fdiff\_9~(k8\_real\_1~(k10\_real\_1~np\_\_3~np\_\_4)~k32\_sin\_cos)=\\
k1\_real\_1~(k7\_square\_1~np\_\_2)){\unicdcdcdhd}(k1\_seq\_1~k1\_fdiff\_9~k32\_sin\_cos=\\
k1\_real\_1~np\_\_1)))\end{array}
\label{hyp:ehddbdpfdhjgogdgpgdhbdad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xreal\_0~X0){\unicdbdjdcd}((k21\_sin\_cos~k6\_numbers=np\_\_1){\unicdcdcdhd}\\
((k18\_sin\_cos~k6\_numbers=k6\_numbers){\unicdcdcdhd}((k20\_sin\_cos~(k4\_xcmplx\_0~\\
X0)=k20\_sin\_cos~X0){\unicdcdcdhd}(k17\_sin\_cos~(k4\_xcmplx\_0~X0)=k4\_xcmplx\_0~\\
(k17\_sin\_cos~X0)))))\end{array}
\label{hyp:ehddbdpfdhjgogpfdgpgdh}
\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}
\begin{array}{c}(m2\_subset\_1~np\_\_0~k1\_numbers~k5\_numbers){\unicdcdcdhd}((m1\_subset\_1~np\_\_0~\\
k5\_numbers){\unicdcdcdhd}(m1\_subset\_1~np\_\_0~k1\_numbers))\end{array}
\label{hyp:dhahdgadpfogfhngfgchbgmgdh}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_xboole\_0}~{np\_\_0}}\label{hyp:dhahdgadpfcgpgpgmgfg}
\end{equation}
Assume the following.
\begin{equation}
{{k6\_numbers}={k1\_xboole\_0}}\label{hyp:chfgegfgggjgogjgehjgpgogpflggdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k3\_sin\_cos6}~{X0}}={{k2\_sin\_cos6}~{X0}}}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfdhjgogpfdgpgdhgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(k3\_rcomp\_1~\\
X0~X1=k2\_xxreal\_1~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgddpfchdgpgngahpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
k2\_rcomp\_1~X0~X1=k4\_xxreal\_1~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfchdgpgngahpfbd}
\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 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(k18\_sin\_cos~X0=k17\_sin\_cos~\\
X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdidpfdhjgogpfdgpgdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}k10\_real\_1~k32\_sin\_cos~np\_\_4 \in k2\_rcomp\_1~(k1\_real\_1~(k10\_real\_1~\\
k32\_sin\_cos~np\_\_2))~(k10\_real\_1~k32\_sin\_cos~np\_\_2)\end{array}
\label{hyp:mgcdedpfdhjgogpfdgpgdhjd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(k6\_numbers \in k3\_rcomp\_1~k6\_numbers~(k10\_real\_1~k32\_sin\_cos~np\_\_2)){\unicdcdcdhd}\\
(k10\_real\_1~k32\_sin\_cos~np\_\_4 \in k3\_rcomp\_1~k6\_numbers~(k10\_real\_1~\\
k32\_sin\_cos~np\_\_2))\end{array}
\label{hyp:mgbdhdpfdhjgogdgpgdhbdad}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v1\_xcmplx\_0}~{X0}})}{\unicdbdjdcd}{({{{k4\_xcmplx\_0}~{({{k4\_xcmplx\_0}~{X0}})}}={X0}})}}}\label{hyp:jgogghpgmgfhehjgghfgogfgdhdhpflgedpfihdgngahmgihpfad}
\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 .  (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}
{{v3\_membered}~{k1\_numbers}}\label{hyp:ggdgddpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k1\_numbers){\unicdbdjdcd}(m1\_subset\_1~(k5\_sin\_cos9~\\
X0)~k1\_numbers)\end{array}
\label{hyp:egehpflgfdpfdhjgogpfdgpgdhjd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{m1\_subset\_1}~{({{k3\_sin\_cos6}~{X0}})}}~{k1\_numbers}}}\label{hyp:egehpflgddpfdhjgogpfdgpgdhgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~k1\_sin\_cos9){\unicdcdcdhd}(m1\_subset\_1~k1\_sin\_cos9~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~k1\_numbers~k1\_numbers)))\end{array}
\label{hyp:egehpflgbdpfdhjgogpfdgpgdhjd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_relat\_1~X0){\unicdcdcdhd}((v1\_funct\_1~X0){\unicdcdcdhd}(v3\_valued\_0~\\
X0))){\unicdbdjdcd}(m1\_subset\_1~(k1\_seq\_1~X0~X1)~k1\_numbers)\end{array}
\label{hyp:egehpflgbdpfdhfgbhpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{{k2\_sin\_cos6}~{X0}}={{{k1\_seq\_1}~{k1\_sin\_cos6}}~{X0}}}}\label{hyp:egcdpfdhjgogpfdgpgdhgd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v1\_xxreal\_0~X0){\unicdcdcdhd}(v1\_xxreal\_0~X1)){\unicdbdjdcd}(\\
(r1\_xxreal\_0~X0~X1){\unicdcdcdid}(r1\_xxreal\_0~X1~X0))\end{array}
\label{hyp:dgpgogogfgdgehfgegogfgdhdhpfchbdpfihihchfgbgmgpfad}
\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}
{{\unicdcdadad} X0 .  {{({{v3\_membered}~{X0}})}{\unicdbdjdcd}{({{v2\_membered}~{X0}})}}}\label{hyp:dgdgedpfngfgngcgfgchfgeg}
\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}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (m1\_subset\_1~X2~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}(v1\_relat\_1~X2)\end{array}
\label{hyp:dgdgbdpfchfgmgdhfgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (v3\_membered~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}(v3\_valued\_0~X2))\end{array}
\label{hyp:dgdgbdidpfghbgmgfhfgegpfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v3\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v1\_xreal\_0~X1))\end{array}
\label{hyp:dgdgbdedpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v2\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v1\_xxreal\_0~X1))\end{array}
\label{hyp:dgdgbdddpfngfgngcgfgchfgeg}
\end{equation}
\begin{theorem}\label{thm:ehbdadpfjgogehfghgchbgid}
$$\begin{array}{c}k3\_sin\_cos6~(k10\_real\_1~(k7\_square\_1~np\_\_2)~np\_\_2)=k10\_real\_1~\\
k32\_sin\_cos~np\_\_4\end{array}$$
\end{theorem}
\end{document}
