\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{l22\_pre\_ff (TMFRnsYAk7LPMyHQ2Z7FREPgGfuUF5iSUSa)}
\maketitle
Let $ v7\_ordinal1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k1\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ np\_\_2 : {\iota}$ be given.
Let $ k4\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_pre\_ff : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ c2\_\_pre\_ff : {\iota}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ k8\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k8\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k3\_finseq\_2 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k12\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k7\_partfun1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k2\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ 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 $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\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 $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k4\_ordinal1 : {\iota}$ be given.
Let $ k2\_xcmplx\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v6\_membered : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{\unicdcdadad} X1 .  {{({{X0} \in {X1}})}{\unicdbdjdcd}{({{{m1\_subset\_1}~{X0}}~{X1}})}}}}\label{hyp:ehbdpfdhfhcgdhfgeh}
\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}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  {\unicdcdadad} X2 .  (((v1\_funct\_1~X1){\unicdcdcdhd}((v1\_funct\_2~\\
X1~k5\_numbers~X0){\unicdcdcdhd}(m1\_subset\_1~X1~(k1\_zfmisc\_1~(k2\_zfmisc\_1~k5\_numbers~\\
X0))))){\unicdcdcdhd}(v7\_ordinal1~X2)){\unicdbdjdcd}(k8\_nat\_1~X0~X1~X2=k1\_funct\_1~X1~X2)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgidpfogbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
{{k5\_numbers}={k4\_ordinal1}}\label{hyp:chfgegfgggjgogjgehjgpgogpflgfdpfogfhngcgfgchdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  (((v1\_funct\_1~X0){\unicdcdcdhd}((v1\_funct\_2~X0~k5\_numbers~\\
(k3\_finseq\_2~k5\_numbers)){\unicdcdcdhd}(m1\_subset\_1~X0~(k1\_zfmisc\_1~(k2\_zfmisc\_1~\\
k5\_numbers~(k3\_finseq\_2~k5\_numbers)))))){\unicdcdcdhd}(m1\_subset\_1~X1~k5\_numbers)){\unicdbdjdcd}\\
(k2\_pre\_ff~X0~X1=k1\_funct\_1~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfahchfgpfgggg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((m1\_subset\_1~X0~k5\_numbers){\unicdcdcdhd}(v7\_ordinal1~\\
X1)){\unicdbdjdcd}(k2\_nat\_1~X0~X1=k2\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgcdpfogbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  {\unicdcdadad} X1 .  ((v7\_ordinal1~X0){\unicdcdcdhd}(m1\_subset\_1~X1~k5\_numbers)){\unicdbdjdcd}\\
(k1\_nat\_1~X0~X1=k2\_xcmplx\_0~X0~X1)\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpflgbdpfogbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_subset\_1~X0~k5\_numbers){\unicdbdjdcd}(({\unicdcdadad} X1 .  (v7\_ordinal1~\\
X1){\unicdbdjdcd}((k1\_nat\_1~X0~np\_\_2=k4\_nat\_1~np\_\_2~X1){\unicdbdjdcd}(k2\_pre\_ff~c2\_\_pre\_ff~\\
(k2\_nat\_1~X0~np\_\_1)=k8\_finseq\_1~k5\_numbers~(k2\_pre\_ff~c2\_\_pre\_ff~\\
X0)~(k12\_finseq\_1~k5\_numbers~(k7\_partfun1~k5\_numbers~(k2\_pre\_ff~\\
c2\_\_pre\_ff~X0)~X1))))){\unicdcdcdhd}({\unicdcdadad} X1 .  (v7\_ordinal1~X1){\unicdbdjdcd}((k1\_nat\_1~\\
X0~np\_\_2=k2\_nat\_1~(k4\_nat\_1~np\_\_2~X1)~np\_\_1){\unicdbdjdcd}(k2\_pre\_ff~c2\_\_pre\_ff~\\
(k2\_nat\_1~X0~np\_\_1)=k8\_finseq\_1~k5\_numbers~(k2\_pre\_ff~c2\_\_pre\_ff~\\
X0)~(k12\_finseq\_1~k5\_numbers~(k2\_nat\_1~(k7\_partfun1~k5\_numbers~\\
(k2\_pre\_ff~c2\_\_pre\_ff~X0)~X1)~(k7\_partfun1~k5\_numbers~(k2\_pre\_ff~\\
c2\_\_pre\_ff~X0)~(k1\_nat\_1~X1~np\_\_1))))))))\end{array}
\label{hyp:mgcdbdpfahchfgpfgggg}
\end{equation}
Assume the following.
\begin{equation}
{{v6\_membered}~{k4\_ordinal1}}\label{hyp:ggdggdpfngfgngcgfgchfgeg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}(v1\_funct\_1~c2\_\_pre\_ff){\unicdcdcdhd}((v1\_funct\_2~c2\_\_pre\_ff~k5\_numbers~\\
(k3\_finseq\_2~k5\_numbers)){\unicdcdcdhd}(m1\_subset\_1~c2\_\_pre\_ff~(k1\_zfmisc\_1~\\
(k2\_zfmisc\_1~k5\_numbers~(k3\_finseq\_2~k5\_numbers)))))\end{array}
\label{hyp:egehpfdgcdpfpfahchfgpfgggg}
\end{equation}
Assume the following.
\begin{equation}
{{\unicdcdadad} X0 .  {{({{v7\_ordinal1}~{X0}})}{\unicdbdjded}{({{X0} \in {k4\_ordinal1}})}}}\label{hyp:egbdcdpfpgchegjgogbgmgbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (v6\_membered~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~X0){\unicdbdjdcd}\\
(v7\_ordinal1~X1))\end{array}
\label{hyp:dgdgbdhdpfngfgngcgfgchfgeg}
\end{equation}
\begin{theorem}\label{thm:mgcdcdpfahchfgpfgggg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (v7\_ordinal1~X0){\unicdbdjdcd}(({\unicdcdadad} X1 .  (v7\_ordinal1~X1){\unicdbdjdcd}(\\
(k1\_nat\_1~X0~np\_\_2=k4\_nat\_1~np\_\_2~X1){\unicdbdjdcd}(k2\_pre\_ff~c2\_\_pre\_ff~(\\
k1\_nat\_1~X0~np\_\_1)=k8\_finseq\_1~k5\_numbers~(k8\_nat\_1~(k3\_finseq\_2~\\
k5\_numbers)~c2\_\_pre\_ff~X0)~(k12\_finseq\_1~k5\_numbers~(k7\_partfun1~\\
k5\_numbers~(k8\_nat\_1~(k3\_finseq\_2~k5\_numbers)~c2\_\_pre\_ff~X0)~\\
X1))))){\unicdcdcdhd}({\unicdcdadad} X1 .  (v7\_ordinal1~X1){\unicdbdjdcd}((k1\_nat\_1~X0~np\_\_2=k2\_nat\_1~\\
(k4\_nat\_1~np\_\_2~X1)~np\_\_1){\unicdbdjdcd}(k2\_pre\_ff~c2\_\_pre\_ff~(k1\_nat\_1~X0~\\
np\_\_1)=k8\_finseq\_1~k5\_numbers~(k8\_nat\_1~(k3\_finseq\_2~k5\_numbers)~\\
c2\_\_pre\_ff~X0)~(k12\_finseq\_1~k5\_numbers~(k2\_nat\_1~(k7\_partfun1~\\
k5\_numbers~(k8\_nat\_1~(k3\_finseq\_2~k5\_numbers)~c2\_\_pre\_ff~X0)~\\
X1)~(k7\_partfun1~k5\_numbers~(k8\_nat\_1~(k3\_finseq\_2~k5\_numbers)~\\
c2\_\_pre\_ff~X0)~(k1\_nat\_1~X1~np\_\_1))))))))\end{array}$$
\end{theorem}
\end{document}
