<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">F</font> be   <a href="realset2.html#NM2">Field</a>;<br/>

<a NAME="E1:38"/>
 ex <font color="Olive">A</font> being non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  ex <font color="Olive">od</font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Olive">A</font> ex <font color="Olive">nd</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">A</font> ex <font color="Olive">om</font> being   <a href="realset1.html#M3">DnT</a> of <font color="Olive">nd</font>,<font color="Olive">A</font> ex <font color="Olive">nm</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Olive">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">nd</font></span><a href="realset1.html#K5">}</a></span> st <br/>( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K1" target="_self">field</a> <font color="Olive">A</font>,<font color="Olive">od</font>,<font color="Olive">om</font>,<font color="Olive">nd</font>,<font color="Olive">nm</font> &amp;  <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Olive">A</font>,<font color="Olive">od</font>,<font color="Olive">nd</font> #) is   <a href="realset2.html#NM1">Group</a> &amp; ( for <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">P</font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Olive">B</font><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">B</font>  st <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Olive">nd</font></span><a href="realset1.html#K5">}</a></span> &amp; <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">nm</font> &amp; <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">om</font> <a href="realset1.html#K4">!</a> <font color="Olive">A</font>,<font color="Olive">nd</font> holds <br/> <a href="rlvect_1.html#G1">LoopStr</a>(# <font color="Olive">B</font>,<font color="Olive">P</font>,<font color="Olive">e</font> #) is   <a href="realset2.html#NM1">Group</a> ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Olive">A</font> holds <br/> ( <font color="Olive">om</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Olive">od</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">od</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">om</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">od</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">od</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> ) ) )
 
<b>by </b><i><a class="txt" href="realset2.html#E12"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">od</font> being   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font>, <font color="Maroon">nd</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">om</font> being    <a href="realset1.html#M3">DnT</a> of <font color="Maroon">nd</font>,<font color="Maroon">A</font>, <font color="Maroon">nm</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">nd</font></span><a href="realset1.html#K5">}</a></span><b> such that </b><br/><a NAME="E3:38"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K1" target="_self">field</a> <font color="Maroon">A</font>,<font color="Maroon">od</font>,<font color="Maroon">om</font>,<font color="Maroon">nd</font>,<font color="Maroon">nm</font>
 <b>and </b><br/><a NAME="E4:38"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">om</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><font color="Maroon">od</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">od</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">om</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">od</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">od</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">om</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>
<a NAME="E5:38"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">od</font> = <font color="Maroon">od</font> as   <a href="binop_1.html#NM2">BinOp</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> ;<br/>
<b>reconsider </b><font color="Maroon">nd</font> = <font color="Maroon">nd</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> <b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">om</font> = <font color="Maroon">om</font> as    <a href="realset1.html#M3">DnT</a> of <font color="Maroon">nd</font>, <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> <b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">nm</font> = <font color="Maroon">nm</font> as    <a href="subset_1.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="realset1.html#K5">{</a><span class="default"><font color="Maroon">nd</font></span><a href="realset1.html#K5">}</a></span> <b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, </i>;<br/>
<a NAME="E10:38"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">od</font> <a href="funct_2.html#R2">=</a>  <a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, , <a class="txt" href="realset2.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E11:38"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K1" target="_self">field</a> <span class="p1">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">om</font>,<font color="Maroon">nd</font>,<font color="Maroon">nm</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, , <a class="txt" href="realset2.html#E13"><i><font color="Green">Lemma51</font></i></a></i>;<br/>
<a NAME="E12:38"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">nd</font> <a href="hidden.html#R1">=</a>  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, , , <a class="txt" href="realset2.html#E14"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">om</font> = <font color="Maroon">om</font> as    <a href="realset1.html#M3">DnT</a> of  <a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font>, <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> <b>by </b><i><a class="txt" href="realset2.html#E19"><i><font color="Green">Lemma58</font></i></a>, , , <a class="txt" href="realset2.html#E14"><i><font color="Green">Lemma52</font></i></a></i>;<br/>
<a NAME="E14:38"/>
<font color="Maroon">om</font> <a href="funct_2.html#R2">=</a>  <a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="realset2.html#E20"><i><font color="Green">Lemma63</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E15:38"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font> holds <br/> ( <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K3" target="_self">odf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">z</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , </i>;<br/>


</div>
