\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\_2Efinite\_\_map\_2EFOLDL2\_\_FUPDATE\_\_LIST (TMFZEgy7v54V3ukhT7d6hDFNqFrsWRcAtc8)}
\maketitle
\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: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: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: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\_2Efinite\_\_map\_2Efmap : {{\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\_2Efinite\_\_map\_2Efmap~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggjgogjgehfgpfpfngbgahpfcdfeggngbgah}
\end{equation}
Let $ ty\_2Epair\_2Eprod : {{\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\_2Epair\_2Eprod~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeahbgjgchpfcdfeahchpgeg}
\end{equation}
Let $ c\_2Efinite\_\_map\_2EFUPDATE : {{\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\_2Efinite\_\_map\_2EFUPDATE~\\
A\_27a~A\_27b \in (((ty\_2Efinite\_\_map\_2Efmap~A\_27a~A\_27b)^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})^{{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeffafeebeeffe}
\end{equation}
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\_2EFOLDL : {{\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\_2Elist\_2EFOLDL~\\
A\_27a~A\_27b \in (((A\_27b^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{A\_27b}})^{{({{({{A\_27b}^{{A\_27a}}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfegepemeeeme}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggjgogjgehfgpfpfngbgahpfcdfegeffafeebeeffepfpfmejedfef}
We define $\mname{c\_2Efinite\_\_map\_2EFUPDATE\_\_LIST}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{{ap}~{({{{c\_2Elist\_2EFOLDL}~{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}~{({{{ty\_2Efinite\_\_map\_2Efmap}~{A\_27a}}~{A\_27b}})}})}}~{({{{c\_2Efinite\_\_map\_2EFUPDATE}~{A\_27a}}~{A\_27b}})}})}}}$.
\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 $ c\_2Elist\_2EMAP2 : {{\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\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}c\_2Elist\_2EMAP2~A\_27a~A\_27b~A\_27c \in ((((ty\_2Elist\_2Elist~\\
A\_27a)^{{({{ty\_2Elist\_2Elist}~{A\_27c}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfenebeafcd}
\end{equation}
Let $ c\_2Elist\_2EFOLDL2 : {{\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\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}c\_2Elist\_2EFOLDL2~A\_27a~A\_27b~A\_27c \in ((((A\_27a^{{({{ty\_2Elist\_2Elist}~{A\_27c}})}})^{{({{ty\_2Elist\_2Elist}~{A\_27b}})}})^{{A\_27a}})^{{({{({{({{A\_27a}^{{A\_27c}}})}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfegepemeeemecd}
\end{equation}
Let $ ty\_2Enum\_2Enum : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Enum\_2Enum}}\label{hyp:ogfgpfehjhpfcdfeogfhngpfcdfeogfhng}
\end{equation}
Let $ c\_2Elist\_2ELENGTH : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Elist\_2ELENGTH~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfemefeoeheefie}
\end{equation}
Let $ c\_2Elist\_2EZIP : {{\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\_2Elist\_2EZIP~\\
A\_27a~A\_27b \in ((ty\_2Elist\_2Elist~(ty\_2Epair\_2Eprod~A\_27a~A\_27b))^{{({{{ty\_2Epair\_2Eprod}~{({{ty\_2Elist\_2Elist}~{A\_27a}})}}~{({{ty\_2Elist\_2Elist}~{A\_27b}})}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfekfjeaf}
\end{equation}
Let $ c\_2Epair\_2EFST : {{\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\_2Epair\_2EFST~\\
A\_27a~A\_27b \in (A\_27a^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfegedfef}
\end{equation}
Let $ c\_2Epair\_2ESND : {{\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\_2Epair\_2ESND~\\
A\_27a~A\_27b \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfedfoeee}
\end{equation}
Let $ c\_2Epair\_2EABS\_\_prod : {{\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\_2Epair\_2EABS\_\_prod~\\
A\_27a~A\_27b \in ((ty\_2Epair\_2Eprod~A\_27a~A\_27b)^{{({{({{2}^{{A\_27b}}})}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeahbgjgchpfcdfebecedfpfpfahchpgeg}
\end{equation}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfepfcdde}
We define $\mname{c\_2Epair\_2E\_2C}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {A\_27a} .  {{\uniddcgcg} V1y \in {A\_27b} .  {({{{ap}~{({{{c\_2Epair\_2EABS\_\_prod}~{A\_27a}}~{A\_27b}})}}~{({{\uniddcgcg} V2a \in {A\_27a} .  {({{\uniddcgcg} V3b \in {A\_27b} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V2a}})}}~{V0x}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27b}})}}~{V3b}})}}~{V1y}})}})}})}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeahbgjgchpfcdfeffoedeffcfcfjf}
We define $\mname{c\_2Epair\_2EUNCURRY}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{({{A\_27c}^{{A\_27b}}})}^{{A\_27a}}})} .  {{\uniddcgcg} V1v \in {({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{{ap}~{V0f}}~{({{{ap}~{({{{c\_2Epair\_2EFST}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}~{({{{ap}~{({{{c\_2Epair\_2ESND}~{A\_27a}}~{A\_27b}})}}~{V1v}})}})}}}}}}$.
\end{definition}
Let $ c\_2Elist\_2EMAP : {{\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\_2Elist\_2EMAP~\\
A\_27a~A\_27b \in (((ty\_2Elist\_2Elist~A\_27b)^{{({{ty\_2Elist\_2Elist}~{A\_27a}})}})^{{({{A\_27b}^{{A\_27a}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfemgjgdhehpfcdfenebeaf}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\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 .  ((V0x=V0x){\unicdbdjded}\\
True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfecffegemepfpfdemebeffdffe}
\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} 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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0l \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V1f \in (A\_27b^{{A\_27a}}) .  \\
((ap~(c\_2Elist\_2ELENGTH~A\_27b)~(ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~\\
V1f)~V0l))=(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemefeoeheefiepfpfnebeaf}
\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} V0l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V1l2 \in (ty\_2Elist\_2Elist~\\
A\_27b) .  (((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l1)=(ap~(c\_2Elist\_2ELENGTH~\\
A\_27b)~V1l2)){\unicdbdjdcd}(((ap~(c\_2Elist\_2ELENGTH~(ty\_2Epair\_2Eprod~A\_27a~\\
A\_27b))~(ap~(c\_2Elist\_2EZIP~A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Elist\_2Elist~A\_27a)~(ty\_2Elist\_2Elist~A\_27b))~V0l1)~V1l2)))=\\
(ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l1)){\unicdcdcdhd}((ap~(c\_2Elist\_2ELENGTH~\\
(ty\_2Epair\_2Eprod~A\_27a~A\_27b))~(ap~(c\_2Elist\_2EZIP~A\_27a~A\_27b)~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Elist\_2Elist~A\_27a)~(ty\_2Elist\_2Elist~\\
A\_27b))~V0l1)~V1l2)))=(ap~(c\_2Elist\_2ELENGTH~A\_27b)~V1l2))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfemefeoeheefiepfpfkfjeaf}
\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} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0l1 \in (\\
ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V1l2 \in (ty\_2Elist\_2Elist~A\_27b) .  \\
({\unicdcdadad} V2f1 \in (A\_27c^{{A\_27a}}) .  ({\unicdcdadad} V3f2 \in (A\_27d^{{A\_27b}}) .  (((\\
ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l1)=(ap~(c\_2Elist\_2ELENGTH~A\_27b)~\\
V1l2)){\unicdbdjdcd}(((ap~(c\_2Elist\_2EZIP~A\_27c~A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Elist\_2Elist~A\_27c)~(ty\_2Elist\_2Elist~A\_27b))~(ap~(ap~(c\_2Elist\_2EMAP~\\
A\_27a~A\_27c)~V2f1)~V0l1))~V1l2))=(ap~(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b)~(ty\_2Epair\_2Eprod~A\_27c~A\_27b))~({\uniddcgcg} V4p \in (ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b) .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27c~A\_27b)~(ap~V2f1~(ap~(\\
c\_2Epair\_2EFST~A\_27a~A\_27b)~V4p)))~(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~\\
V4p))))~(ap~(c\_2Elist\_2EZIP~A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~\\
(ty\_2Elist\_2Elist~A\_27a)~(ty\_2Elist\_2Elist~A\_27b))~V0l1)~V1l2)))){\unicdcdcdhd}\\
((ap~(c\_2Elist\_2EZIP~A\_27a~A\_27d)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Elist\_2Elist~\\
A\_27a)~(ty\_2Elist\_2Elist~A\_27d))~V0l1)~(ap~(ap~(c\_2Elist\_2EMAP~\\
A\_27b~A\_27d)~V3f2)~V1l2)))=(ap~(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b)~(ty\_2Epair\_2Eprod~A\_27a~A\_27d))~({\uniddcgcg} V5p \in (ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b) .  (ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27d)~(ap~(c\_2Epair\_2EFST~\\
A\_27a~A\_27b)~V5p))~(ap~V3f2~(ap~(c\_2Epair\_2ESND~A\_27a~A\_27b)~V5p)))))~\\
(ap~(c\_2Elist\_2EZIP~A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Elist\_2Elist~\\
A\_27a)~(ty\_2Elist\_2Elist~A\_27b))~V0l1)~V1l2))))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfekfjeafpfpfnebeaf}
\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} V0l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V1l2 \in \\
(ty\_2Elist\_2Elist~A\_27b) .  (((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l1)=\\
(ap~(c\_2Elist\_2ELENGTH~A\_27b)~V1l2)){\unicdbdjdcd}({\unicdcdadad} V2f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  \\
((ap~(ap~(ap~(c\_2Elist\_2EMAP2~A\_27c~A\_27a~A\_27b)~V2f)~V0l1)~V1l2)=\\
(ap~(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~A\_27a~A\_27b)~A\_27c)~\\
(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~A\_27c)~V2f))~(ap~(c\_2Elist\_2EZIP~\\
A\_27a~A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Elist\_2Elist~A\_27a)~\\
(ty\_2Elist\_2Elist~A\_27b))~V0l1)~V1l2))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenebeafcdpfpfnebeaf}
\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} V0l1 \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V1l2 \in \\
(ty\_2Elist\_2Elist~A\_27b) .  (((ap~(c\_2Elist\_2ELENGTH~A\_27a)~V0l1)=\\
(ap~(c\_2Elist\_2ELENGTH~A\_27b)~V1l2)){\unicdbdjdcd}({\unicdcdadad} V2f \in (((A\_27c^{{A\_27b}})^{{A\_27a}})^{{A\_27c}}) .  \\
({\unicdcdadad} V3a \in A\_27c .  ((ap~(ap~(ap~(ap~(c\_2Elist\_2EFOLDL2~A\_27c~A\_27a~\\
A\_27b)~V2f)~V3a)~V0l1)~V1l2)=(ap~(ap~(ap~(c\_2Elist\_2EFOLDL~(ty\_2Epair\_2Eprod~\\
A\_27a~A\_27b)~A\_27c)~({\uniddcgcg} V4a \in A\_27c .  (ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~\\
A\_27b~A\_27c)~(ap~V2f~V4a))))~V3a)~(ap~(c\_2Elist\_2EZIP~A\_27a~A\_27b)~\\
(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Elist\_2Elist~A\_27a)~(ty\_2Elist\_2Elist~\\
A\_27b))~V0l1)~V1l2)))))))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfegepemeeemecdpfpfgepemeeeme}
\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} V0ls \in (ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V1f \in (A\_27b^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27a}})}}) .  \\
((ap~(ap~(c\_2Elist\_2EMAP~(ty\_2Epair\_2Eprod~A\_27a~A\_27a)~A\_27b)~\\
V1f)~(ap~(c\_2Elist\_2EZIP~A\_27a~A\_27a)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(\\
ty\_2Elist\_2Elist~A\_27a)~(ty\_2Elist\_2Elist~A\_27a))~V0ls)~V0ls)))=\\
(ap~(ap~(c\_2Elist\_2EMAP~A\_27a~A\_27b)~({\uniddcgcg} V2x \in A\_27a .  (ap~V1f~(ap~\\
(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27a)~V2x)~V2x))))~V0ls))))\end{array}
\label{hyp:ehigngpfcdfemgjgdhehpfcdfenebeafpfpfkfjeafpfpfdfbenefe}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(c\_2Epair\_2EFST~A\_27a~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))=V0x)))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfegedfef}
\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} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(c\_2Epair\_2ESND~A\_27a~\\
A\_27b)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V0x)~V1y))=V1y)))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfedfoeee}
\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} V0f \in ((A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1x \in \\
A\_27a .  ({\unicdcdadad} V2y \in A\_27b .  ((ap~(ap~(\mname{c\_2Epair\_2EUNCURRY}~A\_27a~\\
A\_27b~A\_27c)~V0f)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V1x)~V2y))=\\
(ap~(ap~V0f~V1x)~V2y)))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfeffoedeffcfcfjfpfpfeefege}
\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 (A\_27c^{{({{{ty\_2Epair\_2Eprod}~{A\_27a}}~{A\_27b}})}}) .  \\
(({\uniddcgcg} V1p \in (ty\_2Epair\_2Eprod~A\_27a~A\_27b) .  (ap~V0P~V1p))=(ap~(\\
\mname{c\_2Epair\_2EUNCURRY}~A\_27a~A\_27b~A\_27c)~({\uniddcgcg} V2p1 \in A\_27a .  ({\uniddcgcg} V3p2 \in \\
A\_27b .  (ap~V0P~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27a~A\_27b)~V2p1)~V3p2)))))))\end{array}
\label{hyp:ehigngpfcdfeahbgjgchpfcdfemebeneceeebepfpfafcfpeee}
\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} V0f \in ((A\_27a^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1e \in \\
A\_27a .  ({\unicdcdadad} V2g \in (A\_27b^{{A\_27c}}) .  ({\unicdcdadad} V3l \in (ty\_2Elist\_2Elist~\\
A\_27c) .  ((ap~(ap~(ap~(c\_2Elist\_2EFOLDL~A\_27b~A\_27a)~V0f)~V1e)~(ap~\\
(ap~(c\_2Elist\_2EMAP~A\_27c~A\_27b)~V2g)~V3l))=(ap~(ap~(ap~(c\_2Elist\_2EFOLDL~\\
A\_27c~A\_27a)~({\uniddcgcg} V4x \in A\_27a .  ({\uniddcgcg} V5y \in A\_27c .  (ap~(ap~V0f~V4x)~(\\
ap~V2g~V5y)))))~V1e)~V3l))))))\end{array}
\label{hyp:ehigngpfcdfechjgdgigpfpfmgjgdhehpfcdfegepemeeemepfpfnebeaf}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfeggjgogjgehfgpfpfngbgahpfcdfegepemeeemecdpfpfgeffafeebeeffepfpfmejedfef}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}{\unicdcdadad} A\_27c .  \\
nonempty~A\_27c{\unicdbdjdcd}{\unicdcdadad} A\_27d .  nonempty~A\_27d{\unicdbdjdcd}({\unicdcdadad} V0f1 \in (\\
(A\_27c^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V1f2 \in ((A\_27d^{{A\_27b}})^{{A\_27a}}) .  ({\unicdcdadad} V2bs \in \\
(ty\_2Elist\_2Elist~A\_27a) .  ({\unicdcdadad} V3cs \in (ty\_2Elist\_2Elist~A\_27b) .  \\
({\unicdcdadad} V4a \in (ty\_2Efinite\_\_map\_2Efmap~A\_27c~A\_27d) .  (((ap~(c\_2Elist\_2ELENGTH~\\
A\_27a)~V2bs)=(ap~(c\_2Elist\_2ELENGTH~A\_27b)~V3cs)){\unicdbdjdcd}((ap~(ap~(ap~\\
(ap~(c\_2Elist\_2EFOLDL2~(ty\_2Efinite\_\_map\_2Efmap~A\_27c~A\_27d)~\\
A\_27a~A\_27b)~({\uniddcgcg} V5fm \in (ty\_2Efinite\_\_map\_2Efmap~A\_27c~A\_27d) .  \\
({\uniddcgcg} V6b \in A\_27a .  ({\uniddcgcg} V7c \in A\_27b .  (ap~(ap~(c\_2Efinite\_\_map\_2EFUPDATE~\\
A\_27c~A\_27d)~V5fm)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~A\_27c~A\_27d)~(ap~(ap~V0f1~\\
V6b)~V7c))~(ap~(ap~V1f2~V6b)~V7c)))))))~V4a)~V2bs)~V3cs)=(ap~(ap~(\\
\mname{c\_2Efinite\_\_map\_2EFUPDATE\_\_LIST}~A\_27c~A\_27d)~V4a)~(ap~(c\_2Elist\_2EZIP~\\
A\_27c~A\_27d)~(ap~(ap~(\mname{c\_2Epair\_2E\_2C}~(ty\_2Elist\_2Elist~A\_27c)~\\
(ty\_2Elist\_2Elist~A\_27d))~(ap~(ap~(ap~(c\_2Elist\_2EMAP2~A\_27c~A\_27a~\\
A\_27b)~V0f1)~V2bs)~V3cs))~(ap~(ap~(ap~(c\_2Elist\_2EMAP2~A\_27d~A\_27a~\\
A\_27b)~V1f2)~V2bs)~V3cs)))))))))))\end{array}$$
\end{theorem}
\end{document}
