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

<b>let </b><font color="Maroon">F</font> be   <a href="realset2.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">a</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>, <font color="Maroon">b</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>, <font color="Maroon">c</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>;<br/>



<b>set </b><font color="Maroon">B</font> = <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>;<br/>
<b>set </b><font color="Maroon">P</font> = <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K4">!</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#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">e</font> =  <a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">F</font>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="realset1.html#K5">{</a><span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K4">!</a> <span class="p2">(<span class="default"><a href="realset2.html#K2" target="_self">suppf</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="realset2.html#K6" target="_self">nmf</a> <font color="Maroon">F</font></span>)</span> #) as  <a href="rlvect_1.html#V1">strict</a> <a href="realset2.html#NM1">Group</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font>, <font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">D</font> ;<br/>
<a NAME="E3:43"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="realset1.html#K5">{</a><span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i><a class="ref" href="realset1.html#T11">REALSET1:11</a></i>;<br/>
<a NAME="E4:43"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">s</font>, <font color="Olive">t</font> being   <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span> holds <br/> ( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</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"><font color="Olive">s</font>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>, <font color="Maroon">t</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>;<br/>

<a NAME="E1:43_1"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span><b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="realset2.html#K5" target="_self">omf</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2"><a href="realset1.html#K5">{</a><span class="default"><span class="p3">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="realset2.html#E20"><i><font color="Green">Lemma63</font></i></a></i>;<br/>
<a NAME="E4:43_1"/>
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E5:43_1"/>
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</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"><font color="Maroon">s</font>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:43"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span> holds  <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>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</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>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>, <font color="Maroon">y</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>;<br/>

<a NAME="E1:43_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<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="p3"><a href="realset1.html#K5">{</a><span class="default"><span class="p4">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:43_2"/>
<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="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:43"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span> 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"><span class="p2">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</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> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</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> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</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"><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> <a href="hidden.html#R1">=</a> <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#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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>, <font color="Maroon">y</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>, <font color="Maroon">z</font> be    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>;<br/>

<a NAME="E1:43_3"/><i><font color="Green">E28</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:43_3"/>
<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="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> is    <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"><span class="p2">(<span class="default"><a href="realset2.html#K4" target="_self">ndf</a> <font color="Maroon">F</font></span>)</span></span><a href="realset1.html#K5">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:43_3"/>
( <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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<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="Maroon">y</font>,<font color="Maroon">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#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="Maroon">x</font>,<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="Maroon">y</font>,<font color="Maroon">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><b>end;</b></div>
<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#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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> = 
<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">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<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="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<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="Maroon">a</font>,<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="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E9:43"/>
<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#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="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">c</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#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="Maroon">a</font>,<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="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div>
