\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\_2Edatatype\_\_fp\_\_op (TMJ88x6tqrCiurRSfue2UMDQxk1ywfNxBMb)}
\maketitle
Let $ ty\_2Ebinary\_\_ieee\_2Efp\_\_op : {{\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\_2Efp\_\_op~\\
A0~A1)\end{array}
\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeggahpfpfpgah}
\end{equation}
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\_2Ebinary\_\_ieee\_2Erounding : {\iota}$ be given.
Assume the following.
\begin{equation}
{{nonempty}~{ty\_2Ebinary\_\_ieee\_2Erounding}}\label{hyp:ogfgpfehjhpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfechpgfhogegjgoghg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_MulSub : {{\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\_2EFP\_\_MulSub~\\
A\_27t~A\_27w \in (((((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfnefhmgdffhcg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_MulAdd : {{\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\_2EFP\_\_MulAdd~\\
A\_27t~A\_27w \in (((((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfnefhmgbeegeg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_Div : {{\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\_2EFP\_\_Div~\\
A\_27t~A\_27w \in ((((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfeejggh}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_Mul : {{\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\_2EFP\_\_Mul~\\
A\_27t~A\_27w \in ((((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfnefhmg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_Sub : {{\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\_2EFP\_\_Sub~\\
A\_27t~A\_27w \in ((((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfdffhcg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_Add : {{\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\_2EFP\_\_Add~\\
A\_27t~A\_27w \in ((((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfbeegeg}
\end{equation}
Let $ c\_2Ebinary\_\_ieee\_2EFP\_\_Sqrt : {{\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\_2EFP\_\_Sqrt~\\
A\_27t~A\_27w \in (((ty\_2Ebinary\_\_ieee\_2Efp\_\_op~A\_27t~A\_27w)^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}})^{{ty\_2Ebinary\_\_ieee\_2Erounding}})\end{array}
\label{hyp:ngfgngpfdgpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfegeafpfpfdfbhcheh}
\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}
\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:dgpfcdfecgpgpgmgpfcdfeeebeefbeefjfaffe}
We define $\mname{c\_2Ebool\_2EDATATYPE}$ to be ${{\uniddcgcg} A\_27a : {\iota} .  {({{\uniddcgcg} V0x \in {A\_27a} .  {\mname{c\_2Ebool\_2ET}}})}}$.
\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}
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} V0x \in A\_27a .  ((p~(ap~(\mname{c\_2Ebool\_2EDATATYPE}~\\
A\_27a)~V0x)){\unicdbdjded}True))\end{array}
\label{hyp:ehigngpfcdfecgpgpgmgpfcdfeeebeefbeefjfaffepfpfefbehepfpfefiene}
\end{equation}
\begin{theorem}\label{thm:ehigngpfcdfecgjgogbgchjhpfpfjgfgfgfgpfcdfeegbgehbgehjhahfgpfpfggahpfpfpgah}
$$\begin{array}{c}{\unicdcdadad} A\_27t .  nonempty~A\_27t{\unicdbdjdcd}{\unicdcdadad} A\_27w .  nonempty~A\_27w{\unicdbdjdcd}(\\
{\unicdcdadad} V0fp\_\_op \in (((((((2^{{({{({{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}})^{{({{({{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}})^{{({{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}})^{{({{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}})^{{({{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}})^{{({{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}})^{{({{({{({{{ty\_2Ebinary\_\_ieee\_2Efp\_\_op}~{A\_27t}}~{A\_27w}})}^{{({{{ty\_2Ebinary\_\_ieee\_2Efloat}~{A\_27t}}~{A\_27w}})}}})}^{{ty\_2Ebinary\_\_ieee\_2Erounding}}})}}) .  \\
(p~(ap~(\mname{c\_2Ebool\_2EDATATYPE}~2)~(ap~(ap~(ap~(ap~(ap~(ap~(ap~V0fp\_\_op~\\
(c\_2Ebinary\_\_ieee\_2EFP\_\_Sqrt~A\_27t~A\_27w))~(c\_2Ebinary\_\_ieee\_2EFP\_\_Add~\\
A\_27t~A\_27w))~(c\_2Ebinary\_\_ieee\_2EFP\_\_Sub~A\_27t~A\_27w))~(c\_2Ebinary\_\_ieee\_2EFP\_\_Mul~\\
A\_27t~A\_27w))~(c\_2Ebinary\_\_ieee\_2EFP\_\_Div~A\_27t~A\_27w))~(c\_2Ebinary\_\_ieee\_2EFP\_\_MulAdd~\\
A\_27t~A\_27w))~(c\_2Ebinary\_\_ieee\_2EFP\_\_MulSub~A\_27t~A\_27w)))))\end{array}$$
\end{theorem}
\end{document}
