\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\_\_fn\_\_updates (TMR6YHGja2ATqxCW3uNV85THts5EocXN96c)}
\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\_\_fupd : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R0\_\_fupd~A\_27a \in \\
(((ty\_2Ering\_2Ering~A\_27a)^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})^{{({{A\_27a}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfadpfpfggfhaheg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_R1\_\_fupd : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_R1\_\_fupd~A\_27a \in \\
(((ty\_2Ering\_2Ering~A\_27a)^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})^{{({{A\_27a}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfbdpfpfggfhaheg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RP\_\_fupd : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RP\_\_fupd~A\_27a \in \\
(((ty\_2Ering\_2Ering~A\_27a)^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})^{{({{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfafpfpfggfhaheg}
\end{equation}
Let $ c\_2Ering\_2Ering\_\_RM\_\_fupd : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RM\_\_fupd~A\_27a \in \\
(((ty\_2Ering\_2Ering~A\_27a)^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})^{{({{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfnepfpfggfhaheg}
\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\_\_fupd : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ering\_2Ering\_\_RN\_\_fupd~A\_27a \in \\
(((ty\_2Ering\_2Ering~A\_27a)^{{({{ty\_2Ering\_2Ering}~{A\_27a}})}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{({{A\_27a}^{{A\_27a}}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfechjgoghgpfcdfechjgoghgpfpfcfoepfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f2 \in (A\_27a^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_R0\_\_fupd~A\_27a)~V0f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V1a)~V2a0)~V3f)~V4f0)~\\
V5f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
(ap~V0f2~V1a))~V2a0)~V3f)~V4f0)~V5f1))))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfadpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f2 \in (A\_27a^{{A\_27a}}) .  ({\unicdcdadad} V1a \in \\
A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_R1\_\_fupd~A\_27a)~V0f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V1a)~V2a0)~V3f)~V4f0)~\\
V5f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V1a)~(ap~V0f2~V2a0))~V3f)~V4f0)~V5f1))))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfbdpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f2 \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  \\
({\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_RP\_\_fupd~A\_27a)~V0f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V1a)~V2a0)~V3f)~V4f0)~\\
V5f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V1a)~V2a0)~(ap~V0f2~V3f))~V4f0)~V5f1))))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfafpfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f2 \in (((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  \\
({\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_RM\_\_fupd~A\_27a)~V0f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V1a)~V2a0)~V3f)~V4f0)~\\
V5f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V1a)~V2a0)~V3f)~(ap~V0f2~V4f0))~V5f1))))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfnepfpfggfhaheg}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0f2 \in ((A\_27a^{{A\_27a}})^{{({{A\_27a}^{{A\_27a}}})}}) .  \\
({\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_RN\_\_fupd~A\_27a)~V0f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V1a)~V2a0)~V3f)~V4f0)~\\
V5f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V1a)~V2a0)~V3f)~V4f0)~(ap~V0f2~V5f1)))))))))\end{array}
\label{hyp:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfcfoepfpfggfhaheg}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfechjgoghgpfcdfechjgoghgpfpfggogpfpffhahegbgehfgdh}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}(({\unicdcdadad} V0f2 \in (A\_27a^{{A\_27a}}) .  (\\
{\unicdcdadad} V1a \in A\_27a .  ({\unicdcdadad} V2a0 \in A\_27a .  ({\unicdcdadad} V3f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V4f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V5f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_R0\_\_fupd~A\_27a)~V0f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V1a)~V2a0)~V3f)~V4f0)~\\
V5f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
(ap~V0f2~V1a))~V2a0)~V3f)~V4f0)~V5f1)))))))){\unicdcdcdhd}(({\unicdcdadad} V6f2 \in (A\_27a^{{A\_27a}}) .  \\
({\unicdcdadad} V7a \in A\_27a .  ({\unicdcdadad} V8a0 \in A\_27a .  ({\unicdcdadad} V9f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V10f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V11f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_R1\_\_fupd~A\_27a)~V6f2)~(ap~(ap~(ap~(ap~\\
(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V7a)~V8a0)~V9f)~V10f0)~\\
V11f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~\\
V7a)~(ap~V6f2~V8a0))~V9f)~V10f0)~V11f1)))))))){\unicdcdcdhd}(({\unicdcdadad} V12f2 \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ({\unicdcdadad} V13a \in \\
A\_27a .  ({\unicdcdadad} V14a0 \in A\_27a .  ({\unicdcdadad} V15f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V16f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V17f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_RP\_\_fupd~A\_27a)~V12f2)~(ap~(ap~(ap~(\\
ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V13a)~V14a0)~V15f)~\\
V16f0)~V17f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V13a)~V14a0)~(ap~V12f2~V15f))~V16f0)~V17f1)))))))){\unicdcdcdhd}(({\unicdcdadad} V18f2 \in \\
(((A\_27a^{{A\_27a}})^{{A\_27a}})^{{({{({{A\_27a}^{{A\_27a}}})}^{{A\_27a}}})}}) .  ({\unicdcdadad} V19a \in \\
A\_27a .  ({\unicdcdadad} V20a0 \in A\_27a .  ({\unicdcdadad} V21f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  \\
({\unicdcdadad} V22f0 \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V23f1 \in (A\_27a^{{A\_27a}}) .  \\
((ap~(ap~(c\_2Ering\_2Ering\_\_RM\_\_fupd~A\_27a)~V18f2)~(ap~(ap~(ap~(\\
ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~A\_27a)~V19a)~V20a0)~V21f)~\\
V22f0)~V23f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V19a)~V20a0)~V21f)~(ap~V18f2~V22f0))~V23f1)))))))){\unicdcdcdhd}({\unicdcdadad} V24f2 \in \\
((A\_27a^{{A\_27a}})^{{({{A\_27a}^{{A\_27a}}})}}) .  ({\unicdcdadad} V25a \in A\_27a .  ({\unicdcdadad} V26a0 \in \\
A\_27a .  ({\unicdcdadad} V27f \in ((A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V28f0 \in ((\\
A\_27a^{{A\_27a}})^{{A\_27a}}) .  ({\unicdcdadad} V29f1 \in (A\_27a^{{A\_27a}}) .  ((ap~(ap~(c\_2Ering\_2Ering\_\_RN\_\_fupd~\\
A\_27a)~V24f2)~(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V25a)~V26a0)~V27f)~V28f0)~V29f1))=(ap~(ap~(ap~(ap~(ap~(c\_2Ering\_2Erecordtype\_2Ering~\\
A\_27a)~V25a)~V26a0)~V27f)~V28f0)~(ap~V24f2~V29f1)))))))))))))\end{array}$$
\end{theorem}
\end{document}
