\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\_2Ebinary\_\_ieee\_2Efloat\_\_negate\_\_negate (TMaqDfpxDk6XtEawwMGjFr2HURLYKvMagv4)}
\maketitle
Let $ ty\_2Ebinary\_\_ieee\_2Efloat : {{\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\_2Ebinary\_\_ieee\_2Efloat~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgeh}
\end{equation}
Let $ ty\_2Efcp\_2Ecart : {{\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\_2Efcp\_2Ecart~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfedgbgcheh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27u~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27u}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogegpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  \\
nonempty~A\_27x{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27x)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27x}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogehpfpfggfhaheg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~A\_27t)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogjgggjgdgbgogeg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpffeihahpgogfgogeh}
\end{equation}
Let $ ty\_2Eone\_2Eone : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Eone\_2Eone}}\label{hyp:ogfgpfehjhpfcdfepgogfgpfcdfepgogfg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgog}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd : {{\iota}{\unicdbdjdcd}{{\iota}{\unicdbdjdcd}{\iota}}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdfjghgogpfpfggfhaheg}
\end{equation}
Let $ ty\_2Efcp\_2Efinite\_\_image : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgngbghgfg}
\end{equation}
\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}
Let $ ty\_2Ebool\_2Eitself : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
{{\unicdcdadad} A0 .  {{{nonempty}~{A0}}{\unicdbdjdcd}{{nonempty}~{({{ty\_2Ebool\_2Eitself}~{A0}})}}}}\label{hyp:ogfgpfehjhpfcdfecgpgpgmgpfcdfejgehdhfgmggg}
\end{equation}
Let $ c\_2Ebool\_2Ethe\_\_value : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Ebool\_2Ethe\_\_value~A\_27a \in (\\
ty\_2Ebool\_2Eitself~A\_27a)\end{array}
\label{hyp:ngfgngpfdgpfcdfecgpgpgmgpfcdfeehigfgpfpfghbgmgfhfg}
\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\_2Efcp\_2Edimindex : {{\iota}{\unicdbdjdcd}{\iota}}$ be given.
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}c\_2Efcp\_2Edimindex~A\_27a \in (ty\_2Enum\_2Enum^{{({{ty\_2Ebool\_2Eitself}~{A\_27a}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegjgngjgogegfgih}
\end{equation}
\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}
\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\_2Enum\_2EREP\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EREP\_\_num} \in {({{omega}^{{ty\_2Enum\_2Enum}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfecffeafpfpfogfhng}
\end{equation}
Let $ c\_2Enum\_2ESUC\_\_REP : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2ESUC\_\_REP} \in {({{omega}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfedfffdepfpfcffeaf}
\end{equation}
Let $ c\_2Enum\_2EABS\_\_num : {\iota}$ be given.
Assume the following.
\begin{equation}
{{c\_2Enum\_2EABS\_\_num} \in {({{ty\_2Enum\_2Enum}^{{omega}}})}}\label{hyp:ngfgngpfdgpfcdfeogfhngpfcdfebecedfpfpfogfhng}
\end{equation}
\begin{definition}\label{def:dgpfcdfeogfhngpfcdfedfffde}
We define $\mname{c\_2Enum\_2ESUC}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {({{{ap}~{c\_2Enum\_2EABS\_\_num}}~{({{{ap}~{c\_2Enum\_2ESUC\_\_REP}}~{({{{ap}~{c\_2Enum\_2EREP\_\_num}}~{V0m}})}})}})}}$.
\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:dgpfcdfeahchjgngpfpfchfgdgpfcdfepfddde}
We define $\mname{c\_2Eprim\_\_rec\_2E\_3C}$ to be ${{\uniddcgcg} V0m \in {ty\_2Enum\_2Enum} .  {{\uniddcgcg} V1n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{({{2}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V2P \in {({{2}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V3n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{V2P}}~{({{{ap}~{\mname{c\_2Enum\_2ESUC}}}~{V3n}})}})}})}}~{({{{ap}~{V2P}}~{V3n}})}})}})}})}})}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V2P}}~{V0m}})}})}}~{({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{V2P}}~{V1n}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgpgpgmgpfcdfepfddgepfcdbd}
We define $\mname{c\_2Ebool\_2E\_3F\_21}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0P \in {({{2}^{{A\_27a}}})} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F}}~{A\_27a}})}}~{V0P}})}})}}~{({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V2y \in {A\_27a} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{V0P}}~{V1x}})}})}}~{({{{ap}~{V0P}}~{V2y}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{V1x}})}}~{V2y}})}})}})}})}})}})}})}})}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggjgogjgehfgpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efinite\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{A\_27a}^{{ty\_2Enum\_2Enum}}})}})}}~{({{\uniddcgcg} V0f \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{A\_27a}})}}~{({{\uniddcgcg} V1x \in {A\_27a} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_3F\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2n \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Ebool\_2E\_2F\_5C}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2n}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27a}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27a}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{V0f}}~{V2n}})}})}}~{V1x}})}})}})}})}})}})}})}})}}$.
\end{definition}
Let $ c\_2Efcp\_2Edest\_\_cart : {{\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\_2Efcp\_2Edest\_\_cart~\\
A\_27a~A\_27b \in ((A\_27a^{{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})^{{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})\end{array}
\label{hyp:ngfgngpfdgpfcdfeggdgahpfcdfeegfgdhehpfpfdgbgcheh}
\end{equation}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfeggdgahpfpfjgogegfgih}
We define $\mname{c\_2Efcp\_2Efcp\_\_index}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{({{{c\_2Efcp\_2Edest\_\_cart}~{A\_27a}}~{A\_27b}})}}~{V0x}})}}~{({{{ap}~{({{\mname{c\_2Efcp\_2Efinite\_\_index}}~{({{ty\_2Efcp\_2Efinite\_\_image}~{A\_27b}})}})}}~{V1i}})}})}}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfeggdgahpfcdfegedeaf}
We define $\mname{c\_2Efcp\_2EFCP}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0g \in {({{A\_27a}^{{ty\_2Enum\_2Enum}}})} .  {({{{ap}~{({{\mname{c\_2Emin\_2E\_40}}~{({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})}})}}~{({{\uniddcgcg} V1f \in {({{{ty\_2Efcp\_2Ecart}~{A\_27a}}~{A\_27b}})} .  {({{{ap}~{({{\mname{c\_2Ebool\_2E\_21}}~{ty\_2Enum\_2Enum}})}}~{({{\uniddcgcg} V2i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{({{{ap}~{\mname{c\_2Emin\_2E\_3D\_3D\_3E}}}~{({{{ap}~{({{{ap}~{\mname{c\_2Eprim\_\_rec\_2E\_3C}}}~{V2i}})}}~{({{{ap}~{({{c\_2Efcp\_2Edimindex}~{A\_27b}})}}~{({{c\_2Ebool\_2Ethe\_\_value}~{A\_27b}})}})}})}})}}~{({{{ap}~{({{{ap}~{({{\mname{c\_2Emin\_2E\_3D}}~{A\_27a}})}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{A\_27a}}~{A\_27b}})}}~{V1f}})}}~{V2i}})}})}}~{({{{ap}~{V0g}}~{V2i}})}})}})}})}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfehhpgchegdhpfcdfehhpgchegpfpfbddgpgngah}
We define $\mname{c\_2Ewords\_2Eword\_\_1comp}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} V0w \in {({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27a}})} .  {({{{ap}~{({{{\mname{c\_2Efcp\_2EFCP}}~{2}}~{A\_27a}})}}~{({{\uniddcgcg} V1i \in {ty\_2Enum\_2Enum} .  {({{{ap}~{\mname{c\_2Ebool\_2E\_7E}}}~{({{{ap}~{({{{ap}~{({{{\mname{c\_2Efcp\_2Efcp\_\_index}}~{2}}~{A\_27a}})}}~{V0w}})}}~{V1i}})}})}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfele}
We define $\mname{c\_2Ecombin\_2EK}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {({{\uniddcgcg} V1y \in {A\_27b} .  {V0x}})}})}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfogfghgbgehfg}
We define $\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_negate}$ to be ${{\uniddcgcg} A\_27t : {\iota} .  {{\uniddcgcg} A\_27w : {\iota} .  {{\uniddcgcg} V0x \in {({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})} .  {({{{ap}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd}~{A\_27t}}~{A\_27w}})}}~{({{{ap}~{({{{\mname{c\_2Ecombin\_2EK}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}~{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}})}}~{({{{ap}~{({{\mname{c\_2Ewords\_2Eword\_\_1comp}}~{ty\_2Eone\_2Eone}})}}~{({{{ap}~{({{{c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign}~{A\_27t}}~{A\_27w}})}}~{V0x}})}})}})}})}}~{V0x}})}}}}$.
\end{definition}
\begin{definition}\label{def:dgpfcdfedgpgngcgjgogpfcdfepg}
We define $\mname{c\_2Ecombin\_2Eo}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {{\uniddcgcg} A\_27b : {\iota} .  {{\uniddcgcg} A\_27c : {\iota} .  {{\uniddcgcg} V0f \in {({{A\_27b}^{{A\_27c}}})} .  {{\uniddcgcg} V1g \in {({{A\_27c}^{{A\_27a}}})} .  {({{\uniddcgcg} V2x \in {A\_27a} .  {({{{ap}~{V0f}}~{({{{ap}~{V1g}}~{V2x}})}})}})}}}}}}$.
\end{definition}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27w .  \\
nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  nonempty~A\_27x{\unicdbdjdcd}(({\unicdcdadad} V0f0 \in \\
((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V1f \in \\
(ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27x)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27x)~V0f0)~V1f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V1f)))){\unicdcdcdhd}(({\unicdcdadad} V2f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V3f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27u~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V2f0)~V3f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V3f)))){\unicdcdcdhd}(({\unicdcdadad} V4f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V5f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27t~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~V4f0)~V5f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V5f)))){\unicdcdcdhd}(({\unicdcdadad} V6f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V7f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27u~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V6f0)~V7f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V7f)))){\unicdcdcdhd}(({\unicdcdadad} V8f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V9f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27w)~(ap~(ap~(\\
c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V8f0)~V9f))=\\
(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27w)~V9f)))){\unicdcdcdhd}\\
(({\unicdcdadad} V10f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  \\
({\unicdcdadad} V11f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~A\_27x)~(ap~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~A\_27w~A\_27x)~\\
V10f0)~V11f))=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27t~\\
A\_27w)~V11f)))){\unicdcdcdhd}(({\unicdcdadad} V12f0 \in ((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V13f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27t~A\_27w)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~\\
A\_27t~A\_27w)~V12f0)~V13f))=(ap~V12f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V13f))))){\unicdcdcdhd}(({\unicdcdadad} V14f0 \in ((ty\_2Efcp\_2Ecart~2~\\
A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  ({\unicdcdadad} V15f \in (ty\_2Ebinary\_\_ieee\_2Efloat~\\
A\_27t~A\_27w) .  ((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~A\_27t~\\
A\_27x)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x)~V14f0)~V15f))=(ap~V14f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V15f))))){\unicdcdcdhd}({\unicdcdadad} V16f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V17f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~A\_27u~A\_27w)~(ap~(ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27t~A\_27u~A\_27w)~\\
V16f0)~V17f))=(ap~V16f0~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w)~V17f)))))))))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfbgdgdgggfhahegdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27u .  nonempty~A\_27u{\unicdbdjdcd}{\unicdcdadad} A\_27v .  \\
nonempty~A\_27v{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}{\unicdcdadad} A\_27x .  nonempty~\\
A\_27x{\unicdbdjdcd}{\unicdcdadad} A\_27y .  nonempty~A\_27y{\unicdbdjdcd}(({\unicdcdadad} V0g \in ((ty\_2Efcp\_2Ecart~\\
2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  ({\unicdcdadad} V1f0 \in \\
((ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{ty\_2Eone\_2Eone}})}}) .  \\
({\unicdcdadad} V2f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V1f0)~\\
(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~V0g)~\\
V2f))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign\_\_fupd~A\_27t~A\_27w)~\\
(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)~(\\
ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone)~(ty\_2Efcp\_2Ecart~2~ty\_2Eone\_2Eone))~\\
V1f0)~V0g))~V2f))))){\unicdcdcdhd}(({\unicdcdadad} V3g \in ((ty\_2Efcp\_2Ecart~2~A\_27x)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27w}})}}) .  \\
({\unicdcdadad} V4f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27y)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27x}})}}) .  \\
({\unicdcdadad} V5f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~A\_27x~A\_27y)~\\
V4f0)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~A\_27t~\\
A\_27w~A\_27x)~V3g)~V5f))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent\_\_fupd~\\
A\_27t~A\_27w~A\_27y)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(ty\_2Efcp\_2Ecart~2~\\
A\_27w)~(ty\_2Efcp\_2Ecart~2~A\_27y)~(ty\_2Efcp\_2Ecart~2~A\_27x))~\\
V4f0)~V3g))~V5f))))){\unicdcdcdhd}({\unicdcdadad} V6g \in ((ty\_2Efcp\_2Ecart~2~A\_27u)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27t}})}}) .  \\
({\unicdcdadad} V7f0 \in ((ty\_2Efcp\_2Ecart~2~A\_27v)^{{({{{ty\_2Efcp\_2Ecart}~{2}}~{A\_27u}})}}) .  \\
({\unicdcdadad} V8f \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((ap~(\\
ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~A\_27u~A\_27v~\\
A\_27w)~V7f0)~(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27u~A\_27w)~V6g)~V8f))=(ap~(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand\_\_fupd~\\
A\_27t~A\_27v~A\_27w)~(ap~(ap~(\mname{c\_2Ecombin\_2Eo}~(ty\_2Efcp\_2Ecart~2~\\
A\_27t)~(ty\_2Efcp\_2Ecart~2~A\_27v)~(ty\_2Efcp\_2Ecart~2~A\_27u))~\\
V7f0)~V6g))~V8f)))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfggfhahegggfhahegdh}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
{\unicdcdadad} V0f1 \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ({\unicdcdadad} V1f2 \in \\
(ty\_2Ebinary\_\_ieee\_2Efloat~A\_27t~A\_27w) .  ((V0f1=V1f2){\unicdbdjded}(((ap~\\
(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~A\_27t~A\_27w)~V0f1)=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Sign~\\
A\_27t~A\_27w)~V1f2)){\unicdcdcdhd}(((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V0f1)=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Exponent~\\
A\_27t~A\_27w)~V1f2)){\unicdcdcdhd}((ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w)~V0f1)=(ap~(c\_2Ebinary\_\_ieee\_2Efloat\_\_Significand~\\
A\_27t~A\_27w)~V1f2)))))))\end{array}
\label{hyp:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfdgpgngahpgogfgogehpfpffgbhfhbgmgjgehjh}
\end{equation}
Assume the following.
\begin{equation}
{True}\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeefcfffefie}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0t \in 2 .  (({\unicdcdadad} V1x \in \\
A\_27a .  (p~V0t)){\unicdbdjded}(p~V0t)))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfegepecfbememepfpfdfjeneaf}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}({\unicdcdadad} V0t \in 2 .  (((True{\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}True){\unicdbdjded}\\
(p~V0t)){\unicdcdcdhd}(((False{\unicdcdcdhd}(p~V0t)){\unicdbdjded}False){\unicdcdcdhd}((((p~V0t){\unicdcdcdhd}False){\unicdbdjded}False){\unicdcdcdhd}\\
(((p~V0t){\unicdcdcdhd}(p~V0t)){\unicdbdjded}(p~V0t)))))))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfebeoeeepfpfdemebeffdffedf}
\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} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in A\_27a .  ({\unicdcdadad} V1y \in A\_27b .  ((ap~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27a~A\_27b)~V0x)~V1y)=V0x)))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfefiene}
\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} A\_27e .  nonempty~\\
A\_27e{\unicdbdjdcd}{\unicdcdadad} A\_27f .  nonempty~A\_27f{\unicdbdjdcd}(({\unicdcdadad} V0f \in (A\_27b^{{A\_27a}}) .  \\
({\unicdcdadad} V1v \in A\_27c .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27a~A\_27c~A\_27b)~\\
(ap~(\mname{c\_2Ecombin\_2EK}~A\_27c~A\_27b)~V1v))~V0f)=(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27c~A\_27a)~V1v)))){\unicdcdcdhd}({\unicdcdadad} V2f \in (A\_27e^{{A\_27d}}) .  ({\unicdcdadad} V3v \in \\
A\_27d .  ((ap~(ap~(\mname{c\_2Ecombin\_2Eo}~A\_27f~A\_27e~A\_27d)~V2f)~(ap~(\mname{c\_2Ecombin\_2EK}~\\
A\_27d~A\_27f)~V3v))=(ap~(\mname{c\_2Ecombin\_2EK}~A\_27e~A\_27f)~(ap~V2f~V3v))))))\end{array}
\label{hyp:ehigngpfcdfedgpgngcgjgogpfcdfelepfpfpgpfpfefiene}
\end{equation}
Assume the following.
\begin{equation}
\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}({\unicdcdadad} V0a \in (ty\_2Efcp\_2Ecart~\\
2~A\_27a) .  ((ap~(\mname{c\_2Ewords\_2Eword\_\_1comp}~A\_27a)~(ap~(\mname{c\_2Ewords\_2Eword\_\_1comp}~\\
A\_27a)~V0a))=V0a))\end{array}
\label{hyp:ehigngpfcdfehhpgchegdhpfcdfehfpecfeepfpfoepeefpfpfoepeef}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggmgpgbgehpfpfogfghgbgehfgpfpfogfghgbgehfg}
$$\begin{array}{c}{\unicdcdadad} A\_27a .  nonempty~A\_27a{\unicdbdjdcd}{\unicdcdadad} A\_27b .  nonempty~A\_27b{\unicdbdjdcd}(\\
{\unicdcdadad} V0x \in (ty\_2Ebinary\_\_ieee\_2Efloat~A\_27a~A\_27b) .  ((ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_negate}~\\
A\_27a~A\_27b)~(ap~(\mname{c\_2Ebinary\_\_ieee\_2Efloat\_\_negate}~A\_27a~A\_27b)~\\
V0x))=V0x))\end{array}$$
\end{theorem}
\end{document}
