\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\_comseq\_3 (TMTQt5L7ahUGyyEqBJL9eaNpcEaZQ4pUXPM)}
\maketitle
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k2\_numbers : {\iota}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\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\_comseq\_3 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_comseq\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_comseq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k5\_complex1 : {\iota}$ be given.
Let $ k1\_numbers : {\iota}$ be given.
Let $ v1\_series\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_seq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k6\_numbers : {\iota}$ be given.
Let $ k7\_comseq\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_comseq\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_complex1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_complex1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k3\_series\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k10\_comseq\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k3\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_complex1 : {\iota}$ be given.
Let $ k4\_comseq\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_comseq\_3 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_series\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ v1\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_valued\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
((v1\_series\_1~X0){\unicdbdjdcd}((v2\_comseq\_2~X0){\unicdcdcdhd}(k2\_seq\_2~X0=k6\_numbers)))\end{array}
\label{hyp:ehedpfdhfgchjgfgdhpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
(((v2\_comseq\_2~(k7\_comseq\_3~X0)){\unicdcdcdhd}(v2\_comseq\_2~(k8\_comseq\_3~\\
X0))){\unicdbdjdcd}((v2\_comseq\_2~X0){\unicdcdcdhd}((k3\_complex1~(k3\_comseq\_2~X0)=k2\_seq\_2~\\
(k7\_comseq\_3~X0)){\unicdcdcdhd}(k4\_complex1~(k3\_comseq\_2~X0)=k2\_seq\_2~(k8\_comseq\_3~\\
X0)))))\end{array}
\label{hyp:ehedcdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
((r2\_funct\_2~k5\_numbers~k1\_numbers~(k3\_series\_1~(k7\_comseq\_3~\\
X0))~(k7\_comseq\_3~(k10\_comseq\_3~X0))){\unicdcdcdhd}(r2\_funct\_2~k5\_numbers~\\
k1\_numbers~(k3\_series\_1~(k8\_comseq\_3~X0))~(k8\_comseq\_3~(k10\_comseq\_3~\\
X0))))\end{array}
\label{hyp:ehcdgdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v1\_xcmplx\_0~X0){\unicdbdjdcd}(k2\_xcmplx\_0~(k3\_complex1~X0)~(k3\_xcmplx\_0~\\
(k4\_complex1~X0)~k7\_complex1)=X0)\end{array}
\label{hyp:ehbdddpfdgpgngahmgfgihbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  {\unicdcdadad} X3 .  (((v1\_funct\_1~X2){\unicdcdcdhd}\\
((v1\_funct\_2~X2~X0~X1){\unicdcdcdhd}(m1\_subset\_1~X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
X0~X1))))){\unicdcdcdhd}((v1\_funct\_1~X3){\unicdcdcdhd}((v1\_funct\_2~X3~X0~X1){\unicdcdcdhd}(m1\_subset\_1~\\
X3~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1)))))){\unicdbdjdcd}((r2\_funct\_2~X0~X1~X2~\\
X3){\unicdbdjded}(X2=X3))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfchcdpfggfhogdgehpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
(k8\_comseq\_3~X0=k4\_comseq\_3~X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgidpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
(k7\_comseq\_3~X0=k3\_comseq\_3~X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflghdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
(k10\_comseq\_3~X0=k2\_series\_1~X0)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdadpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}k5\_complex1=k2\_xcmplx\_0~k6\_numbers~(k3\_xcmplx\_0~k6\_numbers~\\
k7\_complex1)\end{array}
\label{hyp:mgbdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
((v1\_comseq\_3~X0){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k2\_numbers)))))){\unicdbdjdcd}((v1\_funct\_1~(k2\_series\_1~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(k2\_series\_1~X0)~k5\_numbers~k2\_numbers){\unicdcdcdhd}(v2\_comseq\_2~\\
(k2\_series\_1~X0))))\end{array}
\label{hyp:ggdggdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
((v2\_comseq\_2~X0){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k2\_numbers)))))){\unicdbdjdcd}((v1\_funct\_1~(k4\_comseq\_3~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(k4\_comseq\_3~X0)~k5\_numbers~k1\_numbers){\unicdcdcdhd}(v2\_comseq\_2~\\
(k4\_comseq\_3~X0))))\end{array}
\label{hyp:ggdgfdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
((v2\_comseq\_2~X0){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~k2\_numbers)))))){\unicdbdjdcd}((v1\_funct\_1~(k3\_comseq\_3~X0)){\unicdcdcdhd}\\
((v1\_funct\_2~(k3\_comseq\_3~X0)~k5\_numbers~k1\_numbers){\unicdcdcdhd}(v2\_comseq\_2~\\
(k3\_comseq\_3~X0))))\end{array}
\label{hyp:ggdgedpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
{{v1\_membered}~{k2\_numbers}}\label{hyp:ggdgbdpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
((v1\_funct\_1~(k8\_comseq\_3~X0)){\unicdcdcdhd}((v1\_funct\_2~(k8\_comseq\_3~X0)~\\
k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~(k8\_comseq\_3~X0)~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~k5\_numbers~k1\_numbers)))))\end{array}
\label{hyp:egehpflgidpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
((v1\_funct\_1~(k7\_comseq\_3~X0)){\unicdcdcdhd}((v1\_funct\_2~(k7\_comseq\_3~X0)~\\
k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~(k7\_comseq\_3~X0)~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~k5\_numbers~k1\_numbers)))))\end{array}
\label{hyp:egehpflghdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_relat\_1~X0){\unicdcdcdhd}((v1\_funct\_1~X0){\unicdcdcdhd}(v1\_valued\_0~X0))){\unicdbdjdcd}\\
((v1\_relat\_1~(k4\_comseq\_3~X0)){\unicdcdcdhd}(v1\_funct\_1~(k4\_comseq\_3~X0)))\end{array}
\label{hyp:egehpflgedpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
((v1\_funct\_1~(k3\_series\_1~X0)){\unicdcdcdhd}((v1\_funct\_2~(k3\_series\_1~X0)~\\
k5\_numbers~k1\_numbers){\unicdcdcdhd}(m1\_subset\_1~(k3\_series\_1~X0)~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~k5\_numbers~k1\_numbers)))))\end{array}
\label{hyp:egehpflgddpfdhfgchjgfgdhpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_relat\_1~X0){\unicdcdcdhd}((v1\_funct\_1~X0){\unicdcdcdhd}(v1\_valued\_0~X0))){\unicdbdjdcd}\\
((v1\_relat\_1~(k3\_comseq\_3~X0)){\unicdcdcdhd}(v1\_funct\_1~(k3\_comseq\_3~X0)))\end{array}
\label{hyp:egehpflgddpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
(m1\_subset\_1~(k3\_comseq\_2~X0)~k2\_numbers)\end{array}
\label{hyp:egehpflgddpfdgpgngdhfgbhpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
((v1\_funct\_1~(k10\_comseq\_3~X0)){\unicdcdcdhd}((v1\_funct\_2~(k10\_comseq\_3~\\
X0)~k5\_numbers~k2\_numbers){\unicdcdcdhd}(m1\_subset\_1~(k10\_comseq\_3~X0)~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~k5\_numbers~k2\_numbers)))))\end{array}
\label{hyp:egehpflgbdadpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k1\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k1\_numbers))))){\unicdbdjdcd}\\
((v1\_series\_1~X0){\unicdbdjded}(v2\_comseq\_2~(k3\_series\_1~X0)))\end{array}
\label{hyp:egcdpfdhfgchjgfgdhpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{{m1\_subset\_1}~{X0}}~{k2\_numbers}})}{\unicdbdjdcd}{({{v1\_xcmplx\_0}~{X0}})}}}\label{hyp:dgdgddpfihdgngahmgihpfad}
\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 .  (m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
k1\_numbers))){\unicdbdjdcd}(((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~\\
k1\_numbers){\unicdcdcdhd}(v1\_series\_1~X0))){\unicdbdjdcd}((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~\\
X0~k5\_numbers~k1\_numbers){\unicdcdcdhd}(v2\_comseq\_2~X0))))\end{array}
\label{hyp:dgdgbdpfdgpgngdhfgbhpfdd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (v1\_membered~X1){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(k1\_zfmisc\_1~(k2\_zfmisc\_1~X0~X1))){\unicdbdjdcd}(v1\_valued\_0~X2))\end{array}
\label{hyp:dgdgbdgdpfghbgmgfhfgegpfad}
\end{equation}
\begin{theorem}\label{thm:ehfdcdpfdgpgngdhfgbhpfdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~k2\_numbers){\unicdcdcdhd}\\
(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~k2\_numbers))))){\unicdbdjdcd}\\
((v1\_comseq\_3~X0){\unicdbdjdcd}((v2\_comseq\_2~X0){\unicdcdcdhd}(k3\_comseq\_2~X0=k5\_complex1)))\end{array}$$
\end{theorem}
\end{document}
