\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{thm\_2Einftree\_2Einftree\_\_nchotomy (TMSCxH5rCXSB4hmoRVxaqR8pqdN4LR4uWcz)}
\maketitle
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddeepfddeepfddfe}
We define $\mname{c\_2Emin\_2E\_3D\_3D\_3E}$ to be ${{\uniddcgcg} P \in {2} .  {{\uniddcgcg} Q \in {2} .  {{inj\_\_o}~{({{{p}~{P}}{\unicdbdjdcd}{{p}~{Q}}})}}}}$ of type ${\iota}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfddee}
We define $\mname{c\_2Emin\_2E\_3D}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} x \in {A} .  {{\uniddcgcg} y \in {A} .  {{inj\_\_o}~{({{x}={y}})}}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfeef}
We define $\mname{c\_2Ebool\_2ET}$ to be ${({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{2}}})}})}}~{({{\uniddcgcg} V0x \in {2} .  {V0x}})}})}}~{({{\uniddcgcg} V1x \in {2} .  {V1x}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdbd}
We define $\mname{c\_2Ebool\_2E\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{2}^{{A\_27a}}})}})}}~{V0P}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepffddepfcdge}
We define $\mname{c\_2Ebool\_2E\_5C\_2F}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{V2t}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}}~{V2t}})}})}})}})}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfege}
We define $\mname{c\_2Ebool\_2EF}$ to be ${({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V0t \in {2} .  {V0t}})}})}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfengjgogpfcdfepfedad}
We define $\mname{c\_2Emin\_2E\_40}$ to be ${{\uniddcgcg} A .  {{\uniddcgcg} P \in {{2}^{{A}}} .  {{\mathbf{if}}~{({{\unicdcdaddd} x \in {A} .  {{p}~{({{{ap}~{P}}~{x}})}}})}~{\mathbf{then}}~{({{the}~{({{\uniddcgcg} x .  {{{x} \in {A}}{\unicdcdcdhd}{{p}~{({{{ap}~{P}}~{x}})}}}})}})}~{\mathbf{else}}~{({{the}~{({{\uniddcgcg} x .  {{x} \in {A}}})}})}}}}$ of type ${{\iota}{\unicdbdjdcd}{\iota}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddge}
We define $\mname{c\_2Ebool\_2E\_3F}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{V0P}}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{V0P}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfhdfe}
We define $\mname{c\_2Ebool\_2E\_7E}$ to be ${({{\uniddcgcg} V0t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}$.
\end{definition}
Let $ ty\_2Elist\_2Elist : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Elist\_2Elist}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfemgjgdhehpfcdfemgjgdheh}
\end{equation}
Let $ c\_2Elist\_2ETL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ETL~A\_27a \in ((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeefme}
\end{equation}
Let $ c\_2Elist\_2EHD : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2EHD~A\_27a \in (A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeieee}
\end{equation}
Let $ ty\_2Esum\_2Esum : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}nonempty~(ty\_2Esum\_2Esum~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfedhfhngpfcdfedhfhng}
\end{equation}
Let $ ty\_2Einftree\_2Einftree : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A0 .  nonempty~A0{\unicdbdjdcd}{\unicdcdadad} A1 .  nonempty~A1{\unicdbdjdcd}{\unicdcdadad} A2 .  nonempty~\\
A2{\unicdbdjdcd}nonempty~(ty\_2Einftree\_2Einftree~A0~A1~A2)\end{array}
\label{hyp:ogfgpfehjhpfcdfejgogggehchfgfgpfcdfejgogggehchfgfg}
\end{equation}
Let $ c\_2Einftree\_2Efrom\_\_inftree : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27d .  \\
nonempty~A\_27d{\unicdbdjdcd}c\_2Einftree\_2Efrom\_\_inftree~A\_27a~A\_27b~A\_27d \in \\
(((ty\_2Esum\_2Esum~A\_27a~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27d}})}})^{{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27b}}~{A\_27d}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogggehchfgfgpfcdfeggchpgngpfpfjgogggehchfgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfcdgepffdde}
We define $\mname{c\_2Ebool\_2E\_2F\_5C}$ to be ${({{\uniddcgcg} V0t1 \in {2} .  {({{\uniddcgcg} V1t2 \in {2} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{2}})}}~{({{\uniddcgcg} V2t \in {2} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V0t1}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{V1t2}})}}~{V2t}})}})}})}}~{V2t}})}})}})}})}})}$.
\end{definition}
Let $ c\_2Esum\_2EABS\_\_sum : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}c\_2Esum\_2EABS\_\_sum~\\
A\_27a~A\_27b \in ((ty\_2Esum\_2Esum~A\_27a~A\_27b)^{{({{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}^{{2}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfedhfhngpfcdfebecedfpfpfdhfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoecf}
We define $\mname{c\_2Esum\_2EINR}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27b} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3y}})}}~{V0e}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{V1b}})}})}})}})}})}})}}}}$.
\end{definition}
Let $ c\_2Elist\_2ENIL : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ENIL~A\_27a \in (ty\_2Elist\_2Elist~\\
A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfeoejeme}
\end{equation}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfedepeoeee}
We define $\mname{c\_2Ebool\_2ECOND}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0t \in {2} .  {({{\uniddcgcg} V1t1 \in {A\_27a} .  {({{\uniddcgcg} V2t2 \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{A\_27a}})}}~{({{\uniddcgcg} V3x \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2ET}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V1t1}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{2}})}}~{V0t}})}}~{\mname{c\_2Ebool\_2EF}}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V3x}})}}~{V2t2}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Einftree\_2Eto\_\_inftree : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27d .  \\
nonempty~A\_27d{\unicdbdjdcd}c\_2Einftree\_2Eto\_\_inftree~A\_27a~A\_27b~A\_27d \in \\
((ty\_2Einftree\_2Einftree~A\_27a~A\_27b~A\_27d)^{{({{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}^{{({{ty\_2Elist\_2Elist}~{A\_27d}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfejgogggehchfgfgpfcdfeehpgpfpfjgogggehchfgfg}
\end{equation}
\begin{definition}\label{def:dgpfcdfejgogggehchfgfgpfcdfejgoeeg}
We define $\mname{c\_2Einftree\_2EiNd}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0b \in {A\_27b} .  {{\uniddcgcg} V1f \in {({{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}^{{A\_27c}}})} .  {({{{ap}~{({{{{c\_2Einftree\_2Eto\_\_inftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}~{({{\uniddcgcg} V2p \in {({{ty\_2Elist\_2Elist}~{A\_27c}})} .  {({{{ap}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Ebool\_2ECOND}}~{({{{ty\_2Esum\_2Esum}~{A\_27a}}~{A\_27b}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{({{ty\_2Elist\_2Elist}~{A\_27c}})}})}}~{V2p}})}}~{({{c\_2Elist\_2ENIL}~{A\_27c}})}})}})}}~{({{{ap}~{({{{\mname{c\_2Esum\_2EINR}}~{A\_27a}}~{A\_27b}})}}~{V0b}})}})}}~{({{{ap}~{({{{ap}~{({{{{c\_2Einftree\_2Efrom\_\_inftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}~{({{{ap}~{V1f}}~{({{{ap}~{({{c\_2Elist\_2EHD}~{A\_27c}})}}~{V2p}})}})}})}}~{({{{ap}~{({{c\_2Elist\_2ETL}~{A\_27c}})}}~{V2p}})}})}})}})}})}}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedhfhngpfcdfejeoeme}
We define $\mname{c\_2Esum\_2EINL}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0e \in {A\_27a} .  {({{{ap}~{({{{c\_2Esum\_2EABS\_\_sum}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V1b \in {2} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{\uniddcgcg} V3y \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2x}})}}~{V0e}})}})}}~{V1b}})}})}})}})}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfejgogggehchfgfgpfcdfejgmegg}
We define $\mname{c\_2Einftree\_2EiLf}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0a \in {A\_27a} .  {({{{ap}~{({{{{c\_2Einftree\_2Eto\_\_inftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}~{({{\uniddcgcg} V1p \in {({{ty\_2Elist\_2Elist}~{A\_27c}})} .  {({{{ap}~{({{{\mname{c\_2Esum\_2EINL}}~{A\_27a}}~{A\_27b}})}}~{V0a}})}})}})}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  (((p~V0t1){\unicdbdjdcd}(p~V1t2)){\unicdbdjdcd}(((p~\\
V1t2){\unicdbdjdcd}(p~V0t1)){\unicdbdjdcd}((p~V0t1){\unicdbdjded}(p~V1t2))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfbeoeefjedfjfnepfpfbeif}
\end{equation}
Assume the following.
\begin{equation}
{({{\unicdcdadad} V0t \in {2} .  {({{False}{\unicdbdjdcd}{({{p}~{V0t}})}})}})}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegebemedfjeefjf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdaddd} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdid}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}True){\unicdbdjded}True){\unicdcdcdhd}\\
(((False{\unicdcdcdid}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdid}False){\unicdbdjded}(p~V0t)){\unicdcdcdhd}(((p~V0t){\unicdcdcdid}\\
(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfepecfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjdcd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}True){\unicdbdjded}\\
True){\unicdcdcdhd}(((False{\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((((p~V0t){\unicdbdjdcd}(p~V0t)){\unicdbdjded}True){\unicdcdcdhd}((\\
(p~V0t){\unicdbdjdcd}False){\unicdbdjded}(\preoh (p~V0t))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in \\
A\_27a .  ((V0x=V1y){\unicdbdjded}(V1y=V0x))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdfjfnepfpffebf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdbdjded}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdbdjded}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdbdjded}(p~V0t)){\unicdbdjded}(\preoh (p~V0t))){\unicdcdcdhd}(((p~V0t){\unicdbdjded}False){\unicdbdjded}(\preoh (\\
p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefebfpfpfdemebeffdffedf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t1 \in 2 .  ({\unicdcdadad} V1t2 \in 2 .  ({\unicdcdadad} V2t3 \in 2 .  (((p~V0t1){\unicdbdjdcd}\\
((p~V1t2){\unicdbdjdcd}(p~V2t3))){\unicdbdjded}(((p~V0t1){\unicdcdcdhd}(p~V1t2)){\unicdbdjdcd}(p~V2t3))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfjeneafpfpfjeoeefcfpe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0x \in 2 .  ({\unicdcdadad} V1x\_27 \in 2 .  ({\unicdcdadad} V2y \in 2 .  ({\unicdcdadad} V3y\_27 \in \\
2 .  ((((p~V0x){\unicdbdjded}(p~V1x\_27)){\unicdcdcdhd}((p~V1x\_27){\unicdbdjdcd}((p~V2y){\unicdbdjded}(p~V3y\_27)))){\unicdbdjdcd}\\
(((p~V0x){\unicdbdjdcd}(p~V2y)){\unicdbdjded}((p~V1x\_27){\unicdbdjdcd}(p~V3y\_27))))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfejeneafpfpfdepeoehe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdaddd} V1x \in \\
A\_27a .  (V1x=V0a)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfefeifjedfefdfpfpfcffegeme}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  (({\unicdcdaddd} V2x \in A\_27a .  ((V2x=V1a){\unicdcdcdhd}(p~(ap~V0P~V2x)))){\unicdbdjded}(p~(\\
ap~V0P~V1a)))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeffoehfjeoeeepfpfefienecd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0a1 \in A\_27a .  ({\unicdcdadad} V1a2 \in A\_27a .  ({\unicdcdadad} V2b1 \in \\
A\_27b .  ({\unicdcdadad} V3f1 \in ((ty\_2Einftree\_2Einftree~A\_27a~A\_27b~A\_27c)^{{A\_27c}}) .  \\
({\unicdcdadad} V4b2 \in A\_27b .  ({\unicdcdadad} V5f2 \in ((ty\_2Einftree\_2Einftree~\\
A\_27a~A\_27b~A\_27c)^{{A\_27c}}) .  ((((ap~(\mname{c\_2Einftree\_2EiLf}~A\_27a~A\_27b~\\
A\_27c)~V0a1)=(ap~(\mname{c\_2Einftree\_2EiLf}~A\_27a~A\_27b~A\_27c)~V1a2)){\unicdbdjded}\\
(V0a1=V1a2)){\unicdcdcdhd}(((ap~(ap~(\mname{c\_2Einftree\_2EiNd}~A\_27a~A\_27b~A\_27c)~\\
V2b1)~V3f1)=(ap~(ap~(\mname{c\_2Einftree\_2EiNd}~A\_27a~A\_27b~A\_27c)~V4b2)~\\
V5f2)){\unicdbdjded}((V2b1=V4b2){\unicdcdcdhd}(V3f1=V5f2))))))))))\end{array}
\label{hyp:ehigngpfcdfejgogggehchfgfgpfcdfejgogggehchfgfgpfpfbdbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1b \in A\_27b .  ({\unicdcdadad} V2f \in \\
((ty\_2Einftree\_2Einftree~A\_27a~A\_27b~A\_27c)^{{A\_27c}}) .  (\preoh ((ap~(\mname{c\_2Einftree\_2EiLf}~\\
A\_27a~A\_27b~A\_27c)~V0a)=(ap~(ap~(\mname{c\_2Einftree\_2EiNd}~A\_27a~A\_27b~\\
A\_27c)~V1b)~V2f))))))\end{array}
\label{hyp:ehigngpfcdfejgogggehchfgfgpfcdfejgogggehchfgfgpfpfegjgdhehjgogdgeh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0P \in (2^{{({{{{ty\_2Einftree\_2Einftree}~{A\_27a}}~{A\_27b}}~{A\_27c}})}}) .  \\
((({\unicdcdadad} V1a \in A\_27a .  (p~(ap~V0P~(ap~(\mname{c\_2Einftree\_2EiLf}~A\_27a~A\_27b~\\
A\_27c)~V1a)))){\unicdcdcdhd}({\unicdcdadad} V2b \in A\_27b .  ({\unicdcdadad} V3f \in ((ty\_2Einftree\_2Einftree~\\
A\_27a~A\_27b~A\_27c)^{{A\_27c}}) .  (({\unicdcdadad} V4d \in A\_27c .  (p~(ap~V0P~(ap~V3f~\\
V4d)))){\unicdbdjdcd}(p~(ap~V0P~(ap~(ap~(\mname{c\_2Einftree\_2EiNd}~A\_27a~A\_27b~A\_27c)~\\
V2b)~V3f))))))){\unicdbdjdcd}({\unicdcdadad} V5t \in (ty\_2Einftree\_2Einftree~A\_27a~\\
A\_27b~A\_27c) .  (p~(ap~V0P~V5t)))))\end{array}
\label{hyp:ehigngpfcdfejgogggehchfgfgpfcdfejgogggehchfgfgpfpfjgogeg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfejgogggehchfgfgpfcdfejgogggehchfgfgpfpfogdgigpgehpgngjh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}({\unicdcdadad} V0t \in (ty\_2Einftree\_2Einftree~A\_27a~A\_27b~\\
A\_27c) .  (({\unicdcdaddd} V1a \in A\_27a .  (V0t=(ap~(\mname{c\_2Einftree\_2EiLf}~A\_27a~\\
A\_27b~A\_27c)~V1a))){\unicdcdcdid}({\unicdcdaddd} V2b \in A\_27b .  ({\unicdcdaddd} V3d \in ((ty\_2Einftree\_2Einftree~\\
A\_27a~A\_27b~A\_27c)^{{A\_27c}}) .  (V0t=(ap~(ap~(\mname{c\_2Einftree\_2EiNd}~A\_27a~\\
A\_27b~A\_27c)~V2b)~V3d))))))\end{array}$$
\end{theorem}
\end{document}
