\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{t48\_cqc\_the3 (TMN43AgXmwf65jxyjvJSNp7JWoW8f7wkDi6)}
\maketitle
Let $ m1\_qc\_lang1 : {{\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 $ r6\_cqc\_the3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ r1\_cqc\_the3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ k6\_cqc\_lang : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ v6\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r5\_cqc\_the3 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k5\_numbers : {\iota}$ be given.
Let $ r1\_xxreal\_0 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ np\_\_1 : {\iota}$ be given.
Let $ v1\_relat\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_funct\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v1\_finseq\_1 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ k3\_finseq\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k1\_funct\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k2\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_qc\_lang1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_nat\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Let $ k11\_cqc\_lang : {{\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~(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 .  (m2\_subset\_1~X3~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}(((r1\_cqc\_the3~X0~X1~X2){\unicdcdcdhd}(r1\_cqc\_the3~X0~\\
X2~X3)){\unicdbdjdcd}(r1\_cqc\_the3~X0~X1~X3)))))\end{array}
\label{hyp:ehgdpfdgbhdgpfehigfgdd}
\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}(((v6\_qc\_lang1~X1~X0){\unicdcdcdhd}(r1\_cqc\_the3~X0~(\\
k6\_cqc\_lang~X0~X1)~(k6\_cqc\_lang~X0~X2))){\unicdbdjdcd}(r1\_cqc\_the3~X0~X2~X1))))\end{array}
\label{hyp:ehedfdpfdgbhdgpfehigfgdd}
\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}(((v6\_qc\_lang1~X1~X0){\unicdcdcdhd}(r1\_cqc\_the3~X0~X1~\\
X2)){\unicdbdjdcd}(r1\_cqc\_the3~X0~(k6\_cqc\_lang~X0~X2)~(k6\_cqc\_lang~X0~X1)))))\end{array}
\label{hyp:ehedddpfdgbhdgpfehigfgdd}
\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}((r6\_cqc\_the3~X0~X1~X2){\unicdbdjdcd}(r5\_cqc\_the3~X0~\\
X1~X2))))\end{array}
\label{hyp:ehddidpfdgbhdgpfehigfgdd}
\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}((r6\_cqc\_the3~X0~X1~X2){\unicdbdjded}((v6\_qc\_lang1~X1~\\
X0){\unicdcdcdhd}({\unicdcdaddd} X3 .  (m1\_subset\_1~X3~k5\_numbers){\unicdcdcdhd}((r1\_xxreal\_0~np\_\_1~\\
X3){\unicdcdcdhd}({\unicdcdaddd} X4 .  ((v1\_relat\_1~X4){\unicdcdcdhd}((v1\_funct\_1~X4){\unicdcdcdhd}(v1\_finseq\_1~\\
X4))){\unicdcdcdhd}((k3\_finseq\_1~X4=X3){\unicdcdcdhd}((k1\_funct\_1~X4~np\_\_1=X2){\unicdcdcdhd}((k1\_funct\_1~\\
X4~X3=X1){\unicdcdcdhd}({\unicdcdadad} X5 .  (m1\_subset\_1~X5~k5\_numbers){\unicdbdjdcd}(\preoh (r1\_xxreal\_0~\\
np\_\_1~X5){\unicdcdcdhd}((\preoh r1\_xxreal\_0~X3~X5){\unicdcdcdhd}({\unicdcdadad} X6 .  (m2\_subset\_1~X6~(\\
k2\_qc\_lang1~X0)~(k3\_qc\_lang1~X0)){\unicdbdjdcd}({\unicdcdadad} X7 .  (m2\_subset\_1~X7~\\
(k9\_qc\_lang1~X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}(\preoh (X7=k1\_funct\_1~X4~X5){\unicdcdcdhd}(\\
k1\_funct\_1~X4~(k2\_nat\_1~X5~np\_\_1)=k11\_cqc\_lang~X0~X6~X7)))))))))))))))))\end{array}
\label{hyp:eggdpfdgbhdgpfehigfgdd}
\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}((r5\_cqc\_the3~X0~X1~X2){\unicdbdjded}((r1\_cqc\_the3~X0~\\
X1~X2){\unicdcdcdhd}(r1\_cqc\_the3~X0~X2~X1)))))\end{array}
\label{hyp:egfdpfdgbhdgpfehigfgdd}
\end{equation}
\begin{theorem}\label{thm:ehedidpfdgbhdgpfehigfgdd}
$$\begin{array}{c}{\unicdcdadad} X0 .  (m1\_qc\_lang1~X0){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_subset\_1~X1~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m2\_subset\_1~X2~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m2\_subset\_1~X3~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m2\_subset\_1~X4~(k9\_qc\_lang1~\\
X0)~(k3\_cqc\_lang~X0)){\unicdbdjdcd}(((r6\_cqc\_the3~X0~X1~X2){\unicdcdcdhd}(r6\_cqc\_the3~X0~\\
X3~X4)){\unicdbdjdcd}((r1\_cqc\_the3~X0~X2~X4){\unicdbdjded}(r1\_cqc\_the3~X0~(k6\_cqc\_lang~X0~\\
X3)~(k6\_cqc\_lang~X0~X1))))))))\end{array}$$
\end{theorem}
\end{document}
