\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{t12\_incproj (TMSqJfcTLncKHSD4yRJaQGdkKYeJEv4JndJ)}
\maketitle
Let $ v2\_struct\_0 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_collsp : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_collsp : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v4\_collsp : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v2\_anproj\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ v3\_anproj\_2 : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ l1\_collsp : {{\iota}{\unicdbdjdcd}{o}}$ be given.
Let $ m1\_subset\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ u2\_incsp\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k3\_incproj : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u1\_incsp\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ r1\_incsp\_1 : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}$ be given.
Let $ u1\_struct\_0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m1\_incproj : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ r1\_collsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}}}$ be given.
Let $ k1\_incproj : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ u3\_incsp\_1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ k2\_incproj : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Let $ m2\_collsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{o}}}$ be given.
Let $ k1\_collsp : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~(u1\_incsp\_1~(k3\_incproj~X0))){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~X2~\\
(u2\_incsp\_1~(k3\_incproj~X0))){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}({\unicdcdadad} X4 .  (m1\_incproj~X4~X0){\unicdbdjdcd}(((X1=X3){\unicdcdcdhd}(X2=X4)){\unicdbdjdcd}((r1\_incsp\_1~\\
(k3\_incproj~X0)~X1~X2){\unicdbdjded}(X3 \in X4)))))))\end{array}
\label{hyp:ehfdpfjgogdgahchpgkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_incproj~\\
X1~X0){\unicdbdjded}(m1\_subset\_1~X1~(u2\_incsp\_1~(k3\_incproj~X0))))\end{array}
\label{hyp:ehedpfjgogdgahchpgkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
(l1\_collsp~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~X0)){\unicdbdjdcd}((r1\_collsp~X0~X1~X2~X3){\unicdbdjdcd}((r1\_collsp~X0~X2~\\
X1~X3){\unicdcdcdhd}(r1\_collsp~X0~X1~X3~X2))))))\end{array}
\label{hyp:ehedpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~\\
X1~(u1\_struct\_0~X0)){\unicdbdjded}(m1\_subset\_1~X1~(u1\_incsp\_1~(k3\_incproj~\\
X0))))\end{array}
\label{hyp:ehddpfjgogdgahchpgkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}((u1\_incsp\_1~(k3\_incproj~\\
X0)=u1\_struct\_0~X0){\unicdcdcdhd}((u2\_incsp\_1~(k3\_incproj~X0)=k1\_incproj~\\
X0){\unicdcdcdhd}(u3\_incsp\_1~(k3\_incproj~X0)=k2\_incproj~X0)))\end{array}
\label{hyp:ehcdpfjgogdgahchpgkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_collsp~X1~\\
X0){\unicdbdjdcd}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdcdcdhd}({\unicdcdaddd} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X0)){\unicdcdcdhd}((X2{\unicdcdgdad}X3){\unicdcdcdhd}((X2 \in X1){\unicdcdcdhd}(X3 \in X1))))))\end{array}
\label{hyp:ehbdedpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
(l1\_collsp~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~\\
X3~(u1\_struct\_0~X0)){\unicdbdjdcd}((r1\_collsp~X0~X1~X2~X3){\unicdbdjded}(X3 \in k1\_collsp~X0~\\
X1~X2)))))\end{array}
\label{hyp:ehbdbdpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
(l1\_collsp~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}((X1 \in k1\_collsp~\\
X0~X1~X2){\unicdcdcdhd}(X2 \in k1\_collsp~X0~X1~X2))))\end{array}
\label{hyp:ehbdadpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_incproj~\\
X1~X0){\unicdbdjded}(m2\_collsp~X1~X0))\end{array}
\label{hyp:chfgegfgggjgogjgehjgpgogpfngbdpfjgogdgahchpgkg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}(l1\_collsp~X0))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m2\_collsp~X1~\\
X0){\unicdbdjded}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdcdcdhd}({\unicdcdaddd} X3 .  \\
(m1\_subset\_1~X3~(u1\_struct\_0~X0)){\unicdcdcdhd}((X2{\unicdcdgdad}X3){\unicdcdcdhd}(X1=k1\_collsp~X0~\\
X2~X3)))))\end{array}
\label{hyp:eghdpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_collsp~X0)){\unicdbdjdcd}((v2\_collsp~X0){\unicdbdjded}\\
({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (m1\_subset\_1~\\
X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~(u1\_struct\_0~\\
X0)){\unicdbdjdcd}((r1\_collsp~X0~X1~X2~X1){\unicdcdcdhd}((r1\_collsp~X0~X1~X1~X2){\unicdcdcdhd}(r1\_collsp~\\
X0~X1~X2~X2)))))))\end{array}
\label{hyp:eghdpfbgogahchpgkgpfcd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_collsp~X0)){\unicdbdjdcd}((v4\_collsp~X0){\unicdbdjded}\\
(\preoh {\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  (\\
m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X3 .  (m1\_subset\_1~X3~\\
(u1\_struct\_0~X0)){\unicdbdjdcd}(r1\_collsp~X0~X1~X2~X3)))))\end{array}
\label{hyp:eggdpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
(l1\_collsp~X0)))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}\\
({\unicdcdadad} X2 .  (m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}(k1\_collsp~X0~X1~\\
X2=ReplSep~(toset~({\uniddcgcg} X3 : \iota .  m1\_subset\_1~X3~(u1\_struct\_0~X0)))~\\
({\uniddcgcg} X3 : \iota .  r1\_collsp~X0~X1~X2~X3)~({\uniddcgcg} X3 : \iota .  X3))))\end{array}
\label{hyp:egfdpfdgpgmgmgdhah}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}(l1\_collsp~X0)){\unicdbdjdcd}((v3\_anproj\_2~\\
X0){\unicdbdjded}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdadad} X2 .  \\
(m1\_subset\_1~X2~(u1\_struct\_0~X0)){\unicdbdjdcd}({\unicdcdaddd} X3 .  (m1\_subset\_1~X3~\\
(u1\_struct\_0~X0)){\unicdcdcdhd}((X1{\unicdcdgdad}X3){\unicdcdcdhd}((X2{\unicdcdgdad}X3){\unicdcdcdhd}(r1\_collsp~X0~X1~X2~X3)))))))\end{array}
\label{hyp:egbdadpfbgogahchpgkgpfcd}
\end{equation}
\begin{theorem}\label{thm:ehbdcdpfjgogdgahchpgkg}
$$\begin{array}{c}{\unicdcdadad} X0 .  ((\preoh v2\_struct\_0~X0){\unicdcdcdhd}((v2\_collsp~X0){\unicdcdcdhd}((v3\_collsp~X0){\unicdcdcdhd}\\
((v4\_collsp~X0){\unicdcdcdhd}((v2\_anproj\_2~X0){\unicdcdcdhd}((v3\_anproj\_2~X0){\unicdcdcdhd}(l1\_collsp~\\
X0))))))){\unicdbdjdcd}({\unicdcdadad} X1 .  (m1\_subset\_1~X1~(u2\_incsp\_1~(k3\_incproj~\\
X0))){\unicdbdjdcd}({\unicdcdaddd} X2 .  (m1\_subset\_1~X2~(u1\_incsp\_1~(k3\_incproj~X0))){\unicdcdcdhd}\\
({\unicdcdaddd} X3 .  (m1\_subset\_1~X3~(u1\_incsp\_1~(k3\_incproj~X0))){\unicdcdcdhd}({\unicdcdaddd} X4 .  \\
(m1\_subset\_1~X4~(u1\_incsp\_1~(k3\_incproj~X0))){\unicdcdcdhd}((X2{\unicdcdgdad}X3){\unicdcdcdhd}((X3{\unicdcdgdad}\\
X4){\unicdcdcdhd}((X4{\unicdcdgdad}X2){\unicdcdcdhd}((r1\_incsp\_1~(k3\_incproj~X0)~X2~X1){\unicdcdcdhd}((r1\_incsp\_1~\\
(k3\_incproj~X0)~X3~X1){\unicdcdcdhd}(r1\_incsp\_1~(k3\_incproj~X0)~X4~X1))))))))))\end{array}$$
\end{theorem}
\end{document}
