\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{t68\_sublemma (TMVr2Vmfvnn4Jn9PC4DLCMhR5yW6F6agM2z)}
\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 $ k9\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_cqc\_lang : {{\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 $ r2\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k5\_relset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k24\_qc\_lang1 : {{\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 $ k2\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k11\_cqc\_lang : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k7\_cqc\_lang : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Let $ k6\_cqc\_lang : {{\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 $ k4\_cqc\_lang : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}}$ be given.
Let $ k5\_cqc\_lang : {{\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~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~X2~(k2\_qc\_lang1~\\
X0)~(k3\_qc\_lang1~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (\preoh v1\_xboole\_0~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(m1\_valuat\_1~X4~X0~X3){\unicdbdjdcd}(({\unicdcdadad} X5 .  (m2\_funct\_2~X5~(k3\_qc\_lang1~\\
X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_funct\_2~X6~(k3\_qc\_lang1~\\
X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~X0)~X3~(\\
k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X5~(k24\_qc\_lang1~X0~X1))~(k5\_relset\_1~\\
(k3\_qc\_lang1~X0)~X3~X6~(k24\_qc\_lang1~X0~X1))){\unicdbdjdcd}((r1\_valuat\_1~X0~\\
X3~X1~X4~X5){\unicdbdjded}(r1\_valuat\_1~X0~X3~X1~X4~X6))))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_funct\_2~\\
X5~(k3\_qc\_lang1~X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_funct\_2~\\
X6~(k3\_qc\_lang1~X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~\\
X0)~X3~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X5~(k24\_qc\_lang1~X0~(k11\_cqc\_lang~\\
X0~X2~X1)))~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X6~(k24\_qc\_lang1~X0~\\
(k11\_cqc\_lang~X0~X2~X1)))){\unicdbdjdcd}((r1\_valuat\_1~X0~X3~(k11\_cqc\_lang~X0~\\
X2~X1)~X4~X5){\unicdbdjded}(r1\_valuat\_1~X0~X3~(k11\_cqc\_lang~X0~X2~X1)~X4~X6))))))))))\end{array}
\label{hyp:ehgdhdpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_subset\_1~X1~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~X2~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (\preoh v1\_xboole\_0~X3){\unicdbdjdcd}({\unicdcdadad} X4 .  \\
(m1\_valuat\_1~X4~X0~X3){\unicdbdjdcd}((({\unicdcdadad} X5 .  (m2\_funct\_2~X5~(k3\_qc\_lang1~\\
X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_funct\_2~X6~(k3\_qc\_lang1~\\
X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~X0)~X3~(\\
k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X5~(k24\_qc\_lang1~X0~X1))~(k5\_relset\_1~\\
(k3\_qc\_lang1~X0)~X3~X6~(k24\_qc\_lang1~X0~X1))){\unicdbdjdcd}((r1\_valuat\_1~X0~\\
X3~X1~X4~X5){\unicdbdjded}(r1\_valuat\_1~X0~X3~X1~X4~X6))))){\unicdcdcdhd}({\unicdcdadad} X5 .  (m2\_funct\_2~\\
X5~(k3\_qc\_lang1~X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_funct\_2~\\
X6~(k3\_qc\_lang1~X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~\\
X0)~X3~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X5~(k24\_qc\_lang1~X0~X2))~\\
(k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X6~(k24\_qc\_lang1~X0~X2))){\unicdbdjdcd}((r1\_valuat\_1~\\
X0~X3~X2~X4~X5){\unicdbdjded}(r1\_valuat\_1~X0~X3~X2~X4~X6)))))){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_funct\_2~\\
X5~(k3\_qc\_lang1~X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}({\unicdcdadad} X6 .  (m2\_funct\_2~\\
X6~(k3\_qc\_lang1~X0)~X3~(k2\_valuat\_1~X0~X3)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~\\
X0)~X3~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X5~(k24\_qc\_lang1~X0~(k7\_cqc\_lang~\\
X0~X1~X2)))~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X3~X6~(k24\_qc\_lang1~X0~\\
(k7\_cqc\_lang~X0~X1~X2)))){\unicdbdjdcd}((r1\_valuat\_1~X0~X3~(k7\_cqc\_lang~X0~X1~\\
X2)~X4~X5){\unicdbdjded}(r1\_valuat\_1~X0~X3~(k7\_cqc\_lang~X0~X1~X2)~X4~X6))))))))))\end{array}
\label{hyp:ehgdcdpfdhfhcgmgfgngngbg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_subset\_1~X1~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (\preoh v1\_xboole\_0~X2){\unicdbdjdcd}({\unicdcdadad} X3 .  \\
(m1\_valuat\_1~X3~X0~X2){\unicdbdjdcd}(({\unicdcdadad} X4 .  (m2\_funct\_2~X4~(k3\_qc\_lang1~\\
X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_funct\_2~X5~(k3\_qc\_lang1~\\
X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~X0)~X2~(\\
k5\_relset\_1~(k3\_qc\_lang1~X0)~X2~X4~(k24\_qc\_lang1~X0~X1))~(k5\_relset\_1~\\
(k3\_qc\_lang1~X0)~X2~X5~(k24\_qc\_lang1~X0~X1))){\unicdbdjdcd}((r1\_valuat\_1~X0~\\
X2~X1~X3~X4){\unicdbdjded}(r1\_valuat\_1~X0~X2~X1~X3~X5))))){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_funct\_2~\\
X4~(k3\_qc\_lang1~X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_funct\_2~\\
X5~(k3\_qc\_lang1~X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~\\
X0)~X2~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X2~X4~(k24\_qc\_lang1~X0~(k6\_cqc\_lang~\\
X0~X1)))~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X2~X5~(k24\_qc\_lang1~X0~(k6\_cqc\_lang~\\
X0~X1)))){\unicdbdjdcd}((r1\_valuat\_1~X0~X2~(k6\_cqc\_lang~X0~X1)~X3~X4){\unicdbdjded}(r1\_valuat\_1~\\
X0~X2~(k6\_cqc\_lang~X0~X1)~X3~X5)))))))))\end{array}
\label{hyp:ehgdbdpfdhfhcgmgfgngngbg}
\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 .  (m2\_funct\_2~\\
X6~(k3\_qc\_lang1~X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m2\_funct\_2~\\
X7~(k3\_qc\_lang1~X0)~X2~(k2\_valuat\_1~X0~X2)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~\\
X0)~X2~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X2~X6~(k24\_qc\_lang1~X0~(k4\_cqc\_lang~\\
X1~X0~X4~X5)))~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X2~X7~(k24\_qc\_lang1~\\
X0~(k4\_cqc\_lang~X1~X0~X4~X5)))){\unicdbdjdcd}((r1\_valuat\_1~X0~X2~(k4\_cqc\_lang~\\
X1~X0~X4~X5)~X3~X6){\unicdbdjded}(r1\_valuat\_1~X0~X2~(k4\_cqc\_lang~X1~X0~X4~X5)~X3~X7))))))))))\end{array}
\label{hyp:ehgdadpfdhfhcgmgfgngngbg}
\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 .  (m2\_funct\_2~X2~(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}\\
({\unicdcdadad} X3 .  (m1\_valuat\_1~X3~X0~X1){\unicdbdjdcd}(r1\_valuat\_1~X0~X1~(k5\_cqc\_lang~\\
X0)~X3~X2))))\end{array}
\label{hyp:ehddcdpfghbgmgfhbgehpfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 : \iota{\unicdbdjdcd}o .  {\unicdcdadad} X1 .  ({\unicdcdadad} X2 .  (m2\_subset\_1~X2~(\\
k9\_qc\_lang1~X1)~(k3\_cqc\_lang~X1)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~\\
(k9\_qc\_lang1~X1)~(k3\_cqc\_lang~X1)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_subset\_1~\\
X4~(k2\_qc\_lang1~X1)~(k3\_qc\_lang1~X1)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m1\_subset\_1~\\
X5~k5\_numbers){\unicdbdjdcd}({\unicdcdadad} X6 .  ((v5\_relat\_1~X6~(k3\_qc\_lang1~X1)){\unicdcdcdhd}\\
((v3\_card\_1~X6~X5){\unicdcdcdhd}(m2\_finseq\_1~X6~(k2\_qc\_lang1~X1)))){\unicdbdjdcd}({\unicdcdadad} X7 .  \\
(m2\_subset\_1~X7~(k6\_qc\_lang1~X1)~(k8\_qc\_lang1~X1~X5)){\unicdbdjdcd}((X0~(k5\_cqc\_lang~\\
X1)){\unicdcdcdhd}((X0~(k4\_cqc\_lang~X5~X1~X7~X6)){\unicdcdcdhd}(((X0~X2){\unicdbdjdcd}(X0~(k6\_cqc\_lang~\\
X1~X2))){\unicdcdcdhd}((((X0~X2){\unicdcdcdhd}(X0~X3)){\unicdbdjdcd}(X0~(k7\_cqc\_lang~X1~X2~X3))){\unicdcdcdhd}((X0~\\
X2){\unicdbdjdcd}(X0~(k11\_cqc\_lang~X1~X4~X2))))))))))))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~\\
X2~(k9\_qc\_lang1~X1)~(k3\_cqc\_lang~X1)){\unicdbdjdcd}(X0~X2))\end{array}
\label{hyp:dhbdpfdgbhdgpfmgbgoghg}
\end{equation}
\begin{theorem}\label{thm:ehgdidpfdhfhcgmgfgngngbg}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (\preoh v1\_xboole\_0~X1){\unicdbdjdcd}(\\
{\unicdcdadad} X2 .  (m1\_valuat\_1~X2~X0~X1){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~\\
(k9\_qc\_lang1~X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_funct\_2~X4~\\
(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}({\unicdcdadad} X5 .  (m2\_funct\_2~\\
X5~(k3\_qc\_lang1~X0)~X1~(k2\_valuat\_1~X0~X1)){\unicdbdjdcd}((r2\_relset\_1~(k3\_qc\_lang1~\\
X0)~X1~(k5\_relset\_1~(k3\_qc\_lang1~X0)~X1~X4~(k24\_qc\_lang1~X0~X3))~\\
(k5\_relset\_1~(k3\_qc\_lang1~X0)~X1~X5~(k24\_qc\_lang1~X0~X3))){\unicdbdjdcd}((r1\_valuat\_1~\\
X0~X1~X3~X2~X4){\unicdbdjded}(r1\_valuat\_1~X0~X1~X3~X2~X5))))))))\end{array}$$
\end{theorem}
\end{document}
