\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\_2Ering\_2Ering\_\_accessors (TMTXMe6wAjj7Ez58QtirTYnftiUGHWcMxGc)}
\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: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 $ ty\_2Ering\_2Ering : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ering\_2Ering}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfechjgoghgpfcdfechjgoghg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R0 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R0~A\_27a \in (A\_27a^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfad}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R1 : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R1~A\_27a \in (A\_27a^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfbd}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RP : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RP~A\_27a \in (((\\
A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfaf}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RM : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RM~A\_27a \in (((\\
A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfne}
\end{equation}
Let $ c\_2Ering\_2Erecordtype\_2Ering : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a \in ((((((ty\_2Ering\_2Ering~A\_27a)^{{({{A\_27a}^{{A\_27a}}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}})^{{A\_27a}})^{{A\_27a}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechfgdgpgchegehjhahfgpfcdfechjgoghg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RN : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RN~A\_27a \in ((A\_27a^{{A\_27a}})^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfoe}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_R0~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V0a))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfad}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_R1~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V1a0))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfbd}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_RP~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V2f))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_RM~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V3f0))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfne}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_RN~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V4f1))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfoe}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfbgdgdgfgdhdhpgchdh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0a \in A\_27a .  ({\unicdcdadad} V1a0 \in \\
A\_27a .  ({\unicdcdadad} V2f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V3f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_R0~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V0a)~V1a0)~\\
V2f)~V3f0)~V4f1))=V0a)))))){\unicdcdcdhd}(({\unicdcdadad} V5a \in A\_27a .  ({\unicdcdadad} V6a0 \in \\
A\_27a .  ({\unicdcdadad} V7f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V8f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V9f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_R1~A\_27a)~\\
(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V5a)~V6a0)~\\
V7f)~V8f0)~V9f1))=V6a0)))))){\unicdcdcdhd}(({\unicdcdadad} V10a \in A\_27a .  ({\unicdcdadad} V11a0 \in \\
A\_27a .  ({\unicdcdadad} V12f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V13f0 \in ((\\
A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V14f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(c\_2Ering\_2Ering\_\_RP~\\
A\_27a)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V10a)~V11a0)~V12f)~V13f0)~V14f1))=V12f)))))){\unicdcdcdhd}(({\unicdcdadad} V15a \in \\
A\_27a .  ({\unicdcdadad} V16a0 \in A\_27a .  ({\unicdcdadad} V17f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V18f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V19f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(c\_2Ering\_2Ering\_\_RM~A\_27a)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V15a)~V16a0)~V17f)~V18f0)~V19f1))=V18f0)))))){\unicdcdcdhd}({\unicdcdadad} V20a \in \\
A\_27a .  ({\unicdcdadad} V21a0 \in A\_27a .  ({\unicdcdadad} V22f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V23f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V24f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(c\_2Ering\_2Ering\_\_RN~A\_27a)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V20a)~V21a0)~V22f)~V23f0)~V24f1))=V24f1))))))))))\end{array}$$
\end{theorem}
\end{document}
