\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{t89\_sublemma (TMHFnytMCCXcNT4v44jQK1zaBexYMjfDvyN)}
\maketitle
Let $ m1\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_xboole\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_valuat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m2\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k16\_substut1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k38\_substut1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m2\_funct\_2 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k3\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_valuat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_valuat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}$ be given.
Let $ k39\_substut1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ r1\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}}$ be given.
Let $ k1\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k3\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k2\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_substut1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k7\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v3\_substut1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k9\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ v2\_substut1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k19\_substut1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k6\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k5\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ k6\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k8\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v5\_relat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ v3\_card\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ m2\_finseq\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_substut1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k4\_sublemma : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_subset\_1~X1~(k2\_qc\_lang1~\\
X0)~(k3\_qc\_lang1~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~X2){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_valuat\_1~X3~X0~X2){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_subset\_1~X4~(k16\_substut1~\\
X0)~(k38\_substut1~X0)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_substut1~X5~X0~(k7\_sublemma~\\
X0~X4~X1)){\unicdbdjdcd}(((v3\_substut1~(k7\_sublemma~X0~X4~X1)~X0){\unicdcdcdhd}({\unicdcdadad} X6 .  \\
(m2\_funct\_2~X6~(k3\_qc\_lang1~X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}((r1\_valuat\_1~\\
X0~X2~(k39\_substut1~X0~X4)~X3~X6){\unicdbdjded}(r1\_sublemma~X0~X4~X2~(k1\_sublemma~\\
X0~X2~X6~(k3\_sublemma~X0~X4~X2~X6))~X3)))){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_funct\_2~\\
X6~(k3\_qc\_lang1~X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}((r1\_valuat\_1~X0~X2~\\
(k39\_substut1~X0~(k9\_sublemma~X0~(k7\_sublemma~X0~X4~X1)~X5))~X3~X6){\unicdbdjded}\\
(r1\_sublemma~X0~(k9\_sublemma~X0~(k7\_sublemma~X0~X4~X1)~X5)~X2~(k1\_sublemma~\\
X0~X2~X6~(k3\_sublemma~X0~(k9\_sublemma~X0~(k7\_sublemma~X0~X4~X1)~X5)~\\
X2~X6))~X3)))))))))\end{array}
\label{hyp:ehididpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  (m1\_valuat\_1~X2~X0~X1){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~\\
(k16\_substut1~X0)~(k38\_substut1~X0)){\unicdbdjdcd}((v2\_substut1~X3~X0){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(m2\_funct\_2~X4~(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~\\
X0~X1~(k39\_substut1~X0~X3)~X2~X4){\unicdbdjded}(r1\_sublemma~X0~X3~X1~(k1\_sublemma~\\
X0~X1~X4~(k3\_sublemma~X0~X3~X1~X4))~X2)))))))\end{array}
\label{hyp:ehedpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  (m1\_valuat\_1~X2~X0~X1){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~\\
(k16\_substut1~X0)~(k38\_substut1~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_subset\_1~\\
X4~(k16\_substut1~X0)~(k38\_substut1~X0)){\unicdbdjdcd}(((k19\_substut1~X0~X3=\\
k19\_substut1~X0~X4){\unicdcdcdhd}(({\unicdcdadad} X5 .  (m2\_funct\_2~X5~(k3\_qc\_lang1~\\
X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~X0~X1~(k39\_substut1~X0~\\
X3)~X2~X5){\unicdbdjded}(r1\_sublemma~X0~X3~X1~(k1\_sublemma~X0~X1~X5~(k3\_sublemma~\\
X0~X3~X1~X5))~X2))){\unicdcdcdhd}({\unicdcdadad} X5 .  (m2\_funct\_2~X5~(k3\_qc\_lang1~X0)~\\
X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~X0~X1~(k39\_substut1~X0~X4)~\\
X2~X5){\unicdbdjded}(r1\_sublemma~X0~X4~X1~(k1\_sublemma~X0~X1~X5~(k3\_sublemma~X0~\\
X4~X1~X5))~X2))))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_funct\_2~X5~(k3\_qc\_lang1~X0)~\\
X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~X0~X1~(k39\_substut1~X0~(k6\_sublemma~\\
X0~X3~X4))~X2~X5){\unicdbdjded}(r1\_sublemma~X0~(k6\_sublemma~X0~X3~X4)~X1~(k1\_sublemma~\\
X0~X1~X5~(k3\_sublemma~X0~(k6\_sublemma~X0~X3~X4)~X1~X5))~X2))))))))\end{array}
\label{hyp:ehcdfdpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  (m1\_valuat\_1~X2~X0~X1){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~\\
(k16\_substut1~X0)~(k38\_substut1~X0)){\unicdbdjdcd}(({\unicdcdadad} X4 .  (m2\_funct\_2~\\
X4~(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~X0~X1~\\
(k39\_substut1~X0~X3)~X2~X4){\unicdbdjded}(r1\_sublemma~X0~X3~X1~(k1\_sublemma~X0~\\
X1~X4~(k3\_sublemma~X0~X3~X1~X4))~X2))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_funct\_2~X4~\\
(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~X0~X1~(k39\_substut1~\\
X0~(k5\_sublemma~X0~X3))~X2~X4){\unicdbdjded}(r1\_sublemma~X0~(k5\_sublemma~X0~X3)~\\
X1~(k1\_sublemma~X0~X1~X4~(k3\_sublemma~X0~(k5\_sublemma~X0~X3)~X1~X4))~\\
X2)))))))\end{array}
\label{hyp:ehbdjdpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~k5\_numbers){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~X2){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_valuat\_1~X3~X0~\\
X2){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_subset\_1~X4~(k6\_qc\_lang1~X0)~(k8\_qc\_lang1~\\
X0~X1)){\unicdbdjdcd}({\unicdcdadad} X5 .  ((v5\_relat\_1~X5~(k3\_qc\_lang1~X0)){\unicdcdcdhd}((v3\_card\_1~\\
X5~X1){\unicdcdcdhd}(m2\_finseq\_1~X5~(k2\_qc\_lang1~X0)))){\unicdbdjdcd}({\unicdcdadad} X6 .  (m1\_subset\_1~\\
X6~(k1\_substut1~X0)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m2\_funct\_2~X7~(k3\_qc\_lang1~\\
X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}((r1\_valuat\_1~X0~X2~(k39\_substut1~X0~\\
(k4\_sublemma~X1~X0~X4~X5~X6))~X3~X7){\unicdbdjded}(r1\_sublemma~X0~(k4\_sublemma~\\
X1~X0~X4~X5~X6)~X2~(k1\_sublemma~X0~X2~X7~(k3\_sublemma~X0~(k4\_sublemma~\\
X1~X0~X4~X5~X6)~X2~X7))~X3)))))))))\end{array}
\label{hyp:ehbdfdpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 : \iota{\unicdbdjdcd}o .  {\unicdcdadad} X1 .  (m1\_qc\_lang1~X1){\unicdbdjdcd}(({\unicdcdadad} X2 .  \\
(m2\_subset\_1~X2~(k16\_substut1~X1)~(k38\_substut1~X1)){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m2\_subset\_1~X3~(k16\_substut1~X1)~(k38\_substut1~X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(m2\_subset\_1~X4~(k2\_qc\_lang1~X1)~(k3\_qc\_lang1~X1)){\unicdbdjdcd}({\unicdcdadad} X5 .  \\
(m1\_substut1~X5~X1~(k7\_sublemma~X1~X2~X4)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m1\_subset\_1~\\
X6~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X7 .  ((v5\_relat\_1~X7~(k3\_qc\_lang1~X1)){\unicdcdcdhd}\\
((v3\_card\_1~X7~X6){\unicdcdcdhd}(m2\_finseq\_1~X7~(k2\_qc\_lang1~X1)))){\unicdbdjdcd}({\unicdcdadad} X8 .  \\
(m2\_subset\_1~X8~(k6\_qc\_lang1~X1)~(k8\_qc\_lang1~X1~X6)){\unicdbdjdcd}({\unicdcdadad} X9 .  \\
(m1\_subset\_1~X9~(k1\_substut1~X1)){\unicdbdjdcd}((X0~(k4\_sublemma~X6~X1~X8~X7~\\
X9)){\unicdcdcdhd}(((v2\_substut1~X2~X1){\unicdbdjdcd}(X0~X2)){\unicdcdcdhd}(((X0~X2){\unicdbdjdcd}(X0~(k5\_sublemma~\\
X1~X2))){\unicdcdcdhd}((((k19\_substut1~X1~X2=k19\_substut1~X1~X3){\unicdcdcdhd}((X0~X2){\unicdcdcdhd}\\
(X0~X3))){\unicdbdjdcd}(X0~(k6\_sublemma~X1~X2~X3))){\unicdcdcdhd}(((v3\_substut1~(k7\_sublemma~\\
X1~X2~X4)~X1){\unicdcdcdhd}(X0~X2)){\unicdbdjdcd}(X0~(k9\_sublemma~X1~(k7\_sublemma~X1~X2~X4)~\\
X5))))))))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~X2~(k16\_substut1~\\
X1)~(k38\_substut1~X1)){\unicdbdjdcd}(X0~X2)))\end{array}
\label{hyp:dhbdpfdhfhcgmgfgngngbg}
\end{equation}
\begin{theorem}\label{thm:ehidjdpfdhfhcgmgfgngngbg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  (m1\_valuat\_1~X2~X0~X1){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~\\
(k16\_substut1~X0)~(k38\_substut1~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_funct\_2~\\
X4~(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r1\_valuat\_1~X0~X1~\\
(k39\_substut1~X0~X3)~X2~X4){\unicdbdjded}(r1\_sublemma~X0~X3~X1~(k1\_sublemma~X0~\\
X1~X4~(k3\_sublemma~X0~X3~X1~X4))~X2))))))\end{array}$$
\end{theorem}
\end{document}
