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

<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">K</font> be  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:109"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">K</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1"/>
( <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_1"><b>suppose </b></a><a NAME="E1:109_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b><font color="Maroon">V</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>;<br/><b>thus </b><a NAME="E2:109_1_1"/>
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="topgrp_1.html#T21">TOPGRP_1:21</a></i>;<br/><a NAME="E3:109_1_1"/>
<font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="rltopsp1.html#T9" target="_self">Th9</a>, <a class="ref" href="convex1.html#T40">CONVEX1:40</a></i>;<br/><b>hence </b><a NAME="E4:109_1_1"/>
<font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_2"><b>suppose </b></a><a NAME="E1:109_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><i><font color="Green">E48</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:109_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:109_1_2_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</a></i>;<br/><a NAME="E3:109_1_2_1"/>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="rltopsp1.html#T8" target="_self">Th8</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/><a NAME="E4:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, </i>;<br/><a NAME="E5:109_1_2_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>;<br/><a NAME="E6:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/><b>then consider </b><font color="Maroon">V'</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:109_1_2_1"/>
<font color="Maroon">V'</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>and </b><br/><a NAME="E9:109_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">V'</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V'</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:109_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><b>consider </b><font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:109_1_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>and </b><br/><a NAME="E12:109_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V'</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:109_1_2"><i><font color="Green">E67</font></i></a></i>;<br/><b>take </b><font color="Maroon">Vx</font> = <font color="Maroon">Vx</font>;<br/><b>thus </b><a NAME="E13:109_1_2_1"/>
<font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>by </b><i/>;<br/><a NAME="E14:109_1_2_1"/>
<font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:109_1_2_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E4:109_1_2_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:109_1_2_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<b>hence </b><a NAME="E6:109_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V'</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E15:109_1_2_1"/><b>then </b>
<font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V'</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E16:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E17:109_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">Vx</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E19:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="convex1.html#T36">CONVEX1:36</a></i>;<br/><a NAME="E20:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E21:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rusub_4.html#K5">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D10">RLVECT_1:def 10</a></i>;<br/><a NAME="E22:109_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E23:109_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">xV</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:109_1_2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:109_1_2_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>and </b><br/><a NAME="E6:109_1_2_2"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a>, </i>;<br/><b>take </b><font color="Maroon">z</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>;<br/><b>thus </b><a NAME="E7:109_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">xV</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">v1</font>, <font color="Olive">v2</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">V1</font>, <font color="Olive">V2</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">v1</font>,<font color="Olive">V1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">v2</font>,<font color="Olive">V2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">v1</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">V1</font> <a href="hidden.html#R1">=</a> <font color="Olive">v2</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">V2</font> );<br/><a NAME="E5:109_1_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:109_1_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 ;<br/>

<a NAME="E2:109_1_2_3"/><b>then </b>
 ex <font color="Olive">v</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">V</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">v</font>,<font color="Olive">V</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">V</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">v</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">V</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div><a NAME="E6:109_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 ;<br/><a NAME="E7:109_1_2"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>given </b><font color="Maroon">v1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">V1</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>, <font color="Maroon">V2</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><a NAME="E1:109_1_2_4"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">V1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E2:109_1_2_4"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">V2</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E3:109_1_2_4"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">v1</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V2</font>
 ;<br/>

<b>given </b><font color="Maroon">w1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">w2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">W1</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>, <font color="Maroon">W2</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><a NAME="E4:109_1_2_4"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w1</font>,<font color="Maroon">W1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E5:109_1_2_4"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">W2</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E6:109_1_2_4"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">w1</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">W2</font>
 ;<br/>

<b>take </b>
<font color="Maroon">v1</font>
;<br/>

<b>take </b>
<font color="Maroon">w2</font>
;<br/>

<b>take </b>
<font color="Maroon">V1</font>
;<br/>

<b>take </b>
<font color="Maroon">W2</font>
;<br/>

<a NAME="E7:109_1_2_4"/>
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">V2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:109_1_2_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">V1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w2</font>,<font color="Maroon">W2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">v1</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">W2</font> )
 <b>by </b><i>, , <a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, <a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">eqR</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">xV</font><b> such that </b><br/><a NAME="E9:109_1_2"/><i><font color="Green">E177</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <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#R2">in</a> <font color="Maroon">eqR</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1">EQREL_1:sch 1</a>(, <a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:109_1_2_5"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">eqR</font>
 ;<br/><a NAME="E2:109_1_2_5"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Olive">x</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:109_1_2_5"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E12:109_1_2"/><i><font color="Green">E179</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">eqR</font>
 <b>and </b><br/><a NAME="E13:109_1_2"/><i><font color="Green">E180</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">eqR</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 <b>by </b><i><a class="ref" href="wellord2.html#T28">WELLORD2:28</a></i>;<br/><a NAME="E14:109_1_2"/><i><font color="Green">E181</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xV</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_6"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:109_1_2_6"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xV</font> )
 
<b>by </b><i>, , <a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E5:109_1_2_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 ;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">xVV</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>;<br/><b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> </span>}</span> ;<br/><a NAME="E15:109_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_7"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_7"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_7"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><a NAME="E17:109_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">F</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a><br/>

<b>assume </b><a NAME="E1:109_1_2_8"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:109_1_2_8"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>and </b><br/><a NAME="E5:109_1_2_8"/><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a>, <a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E6:109_1_2_8"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6">CONNSP_2:6</a></i>;<br/>
<a NAME="E7:109_1_2_8"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:109_1_2_8"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<a NAME="E9:109_1_2_8"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, </i>;<br/>
<a NAME="E10:109_1_2_8"/><b>then </b><i><font color="Green">E189</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">eqR</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E11:109_1_2_8"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:109_1_2_8"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p4"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eqR</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">v1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">V1</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>, <font color="Maroon">V2</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E14:109_1_2_8"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v1</font>,<font color="Maroon">V1</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E15:109_1_2_8"/><i><font color="Green">E191</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v2</font>,<font color="Maroon">V2</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E16:109_1_2_8"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">v1</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">V2</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:109_1_2_8"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">Vx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E18:109_1_2_8"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:109_1_2_8"/><b>then </b>
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a></i>;<br/>
<b>hence </b><a NAME="E20:109_1_2_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:109_1_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:109_1_2_9"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:109_1_2_9"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_9"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E20:109_1_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E21:109_1_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">G</font> <a href="compts_1.html#R1">is_a_cover_of</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E22:109_1_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, <a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/><a NAME="E23:109_1_2"/><i><font color="Green">E198</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:109_1_2_10"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:109_1_2_10"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a></i>;<br/>
<b>hence </b><a NAME="E3:109_1_2_10"/>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 ;<br/>


</div><b>end;</b></div><a NAME="E24:109_1_2"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Vx</font> be  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:109_1_2_11"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E2:109_1_2_11"/><i><font color="Green">E201</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:109_1_2_11"/>
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, <a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vv</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:109_1_2_11"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vv</font>
 <b>and </b><br/><a NAME="E6:109_1_2_11"/><i><font color="Green">E204</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>
<a NAME="E7:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vu</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:109_1_2_11"/><i><font color="Green">E206</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">Vu</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E10:109_1_2_11"/><i><font color="Green">E207</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">Vu</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vu</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vu</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vu</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/>
<a NAME="E11:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eqR</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a>, <a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a>, , <a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a></i>;<br/>
<a NAME="E12:109_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E13:109_1_2_11"/><b>then </b><i><font color="Green">E208</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:109_1_2_11"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E16:109_1_2_11"/><i><font color="Green">E210</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:109_1_2_11"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E19:109_1_2_11"/><i><font color="Green">E212</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:109_1_2_11"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 <b>and </b><br/><a NAME="E22:109_1_2_11"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E24:109_1_2_11"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 <b>and </b><br/><a NAME="E25:109_1_2_11"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E26:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , , <a class="ref" href="rltopsp1.html#D3" target="_self">Def3</a>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E27:109_1_2_11"/><b>then </b><i><font color="Green">E217</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E28:109_1_2_11"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a>, , <a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a></i>;<br/>
<a NAME="E29:109_1_2_11"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T21" target="_self">Th21</a>, <a class="ref" href="rltopsp1.html#D3" target="_self">Def3</a>, , <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a>, <a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E30:109_1_2_11"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">Vx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vv</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">Vv</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vu</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E31:109_1_2_11"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">FVx</font> = <span class="p1">{<span class="default"> <font color="Olive">Vx</font> where <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> :  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/><a NAME="E25:109_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">Vx</font> where <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> :  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_12"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">Vx</font> where <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> :  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_12"/><b>then </b>
 ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">Vx</font> &amp;  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_12"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">FVx</font> = <span class="p1">{<span class="default"> <font color="Olive">Vx</font> where <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> :  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><b>take </b><font color="Maroon">V</font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">FVx</font>;<br/><a NAME="E27:109_1_2"/><i><font color="Green">E219</font></i>: 
<font color="Maroon">FVx</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:109_1_2_13"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 ;<br/>

<a NAME="E2:109_1_2_13"/><b>then </b>
 ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Olive">Vx</font> &amp;  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_13"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


</div><b>end;</b></div><div><i><font color="Green">E220</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_14"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:109_1_2_14"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T9" target="_self">Th9</a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:109_1_2_14"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E5:109_1_2_14"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a>, </i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:109_1_2_14"/><i><font color="Green">E223</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E8:109_1_2_14"/>
<font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E9:109_1_2_14"/>
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:109_1_2_15"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:109_1_2_15"/><i><font color="Green">E225</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><a NAME="E4:109_1_2_15"/>
 ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">Vx</font> &amp;  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) )
 <b>by </b><i><a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a></i>;<br/><a NAME="E5:109_1_2_15"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A'</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><b>hence </b><a NAME="E6:109_1_2_15"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div><a NAME="E30:109_1_2"/><b>then </b><i><font color="Green">E226</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ex <font color="Olive">Vx</font> being <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> &amp; <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Vx</font> );<br/><a NAME="E31:109_1_2"/><i><font color="Green">E227</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:109_1_2_16"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:109_1_2_16"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vz</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:109_1_2_16"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vz</font>
 <b>and </b><br/><a NAME="E5:109_1_2_16"/><i><font color="Green">E231</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">Vz</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>
<b>take </b>
<font color="Maroon">Vz</font>
;<br/>

<b>thus </b><a NAME="E6:109_1_2_16"/>
( <font color="Maroon">Vz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">Vz</font>] )
 <b>by </b><i>, , <a class="ref" href="rltopsp1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">G</font>,<font color="Maroon">FVx</font><b> such that </b><br/><a NAME="E33:109_1_2"/><i><font color="Green">E233</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="ref" href="rltopsp1.html#T20" target="_self">Th20</a>);<br/></i><a NAME="E34:109_1_2"/><i><font color="Green">E234</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E21"><i><font color="Green">Lemma87</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:109_1_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">FVx</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_17"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:109_1_2_17"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vx</font>
 <b>and </b><br/><a NAME="E4:109_1_2_17"/><i><font color="Green">E237</font></i>: 
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:109_1_2_17"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E7:109_1_2_17"/><i><font color="Green">E239</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vv</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:109_1_2_17"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vv</font>
 <b>and </b><br/><a NAME="E10:109_1_2_17"/>
<font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vv</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E11:109_1_2_17"/><i><font color="Green">E241</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E12:109_1_2_17"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Vv</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E13:109_1_2_17"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">eqR</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, , , <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E14:109_1_2_17"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/>
<a NAME="E15:109_1_2_17"/><b>then </b><i><font color="Green">E243</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:109_1_2_17"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E18:109_1_2_17"/><i><font color="Green">E245</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:109_1_2_17"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E21:109_1_2_17"/><i><font color="Green">E247</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E23:109_1_2_17"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 <b>and </b><br/><a NAME="E24:109_1_2_17"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:109_1_2_17"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">xV</font>
 <b>and </b><br/><a NAME="E27:109_1_2_17"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E40"><i><font color="Green">Lemma103</font></i></a>, <a class="ref" href="eqrel_1.html#D5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E28:109_1_2_17"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a>, <a class="ref" href="rltopsp1.html#T28" target="_self">Th28</a>, <a class="txt" href="rltopsp1.html#E42"><i><font color="Green">Lemma106</font></i></a></i>;<br/>
<a NAME="E29:109_1_2_17"/><b>then </b><i><font color="Green">E252</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T29" target="_self">Th29</a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E30:109_1_2_17"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <font color="Maroon">eqR</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="txt" href="rltopsp1.html#E40"><i><font color="Green">Lemma103</font></i></a>, , <a class="ref" href="rltopsp1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E31:109_1_2_17"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">v</font>,<font color="Maroon">Vv</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="rltopsp1.html#T28" target="_self">Th28</a>, , <a class="txt" href="rltopsp1.html#E42"><i><font color="Green">Lemma106</font></i></a>, <a class="ref" href="rltopsp1.html#D7" target="_self">Def7</a>, , <a class="txt" href="rltopsp1.html#E43"><i><font color="Green">Lemma107</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/>
<a NAME="E32:109_1_2_17"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">Vx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vv</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E33:109_1_2_17"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a>, <a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:109_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">FVx</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E37:109_1_2"/><b>then </b>
<font color="Maroon">FVx</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T21" target="_self">Th21</a>, <a class="ref" href="finset_1.html#T27">FINSET_1:27</a></i>;<br/><a NAME="E38:109_1_2"/><b>then </b>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/><a NAME="E39:109_1_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><b>hence </b><a NAME="E40:109_1_2"/>
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T19" target="_self">Th19</a>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><b>set </b><font color="Maroon">FxVxV</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">FxVxVx</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> ;<br/><a NAME="E41:109_1_2"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_18"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:109_1_2_18"/>
<font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:109_1_2_18"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:109_1_2_18"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E6:109_1_2_18"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">Vu</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:109_1_2_18"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vu</font>
 <b>and </b><br/><a NAME="E9:109_1_2_18"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">Vu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:109_1_2_18"/><i><font color="Green">E262</font></i>: 
( <font color="Maroon">Vu</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E12:109_1_2_18"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a>, <a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a>, </i>;<br/>
<a NAME="E13:109_1_2_18"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a>, <a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:109_1_2_18"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E42:109_1_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:109_1_2_19"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_19"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:109_1_2_19"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:109_1_2_19"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E7:109_1_2_19"/><i><font color="Green">E267</font></i>: 
( <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E8:109_1_2_19"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:109_1_2_19"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E11:109_1_2_19"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font>
 <b>and </b><br/><a NAME="E12:109_1_2_19"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D3" target="_self">Def3</a>, </i>;<br/>
<a NAME="E13:109_1_2_19"/>
<font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E14:109_1_2_19"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E15:109_1_2_19"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a>, </i>;<br/>
<a NAME="E16:109_1_2_19"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E17:109_1_2_19"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:109_1_2"/><b>then </b><i><font color="Green">E272</font></i>: 
<font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_20"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:109_1_2_20_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:109_1_2_20_1"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:109_1_2_20_1"/><i><font color="Green">E274</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T9" target="_self">Th9</a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:109_1_2_20_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E8:109_1_2_20_1"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:109_1_2_20_1"/><i><font color="Green">E276</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E11:109_1_2_20_1"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E12:109_1_2_20_1"/><i><font color="Green">E278</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">A</font> = <font color="Maroon">A</font>;<br/><b>take </b><font color="Maroon">B</font> = <font color="Maroon">B</font>;<br/><b>thus </b><a NAME="E14:109_1_2_20_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E15:109_1_2_20_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_20_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_2_20_1_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_20_1_1"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:109_1_2_20_1_1"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E6:109_1_2_20_1_1"/>
<font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:109_1_2_20_1_1"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E9:109_1_2_20_1_1"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E10:109_1_2_20_1_1"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E11:109_1_2_20_1_1"/>
<font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E12:109_1_2_20_1_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T29" target="_self">Th29</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E13:109_1_2_20_1_1"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a>, </i>;<br/><a NAME="E14:109_1_2_20_1_1"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T28" target="_self">Th28</a>, </i>;<br/><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> = 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rltopsp1.html#E40"><i><font color="Green">Lemma103</font></i></a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E16:109_1_2_20_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a>, , <a class="txt" href="rltopsp1.html#E42"><i><font color="Green">Lemma106</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:109_1_2_20_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:109_1_2_20"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E3:109_1_2_20"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:109_1_2_20"/><i><font color="Green">E287</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E5:109_1_2_20"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T25" target="_self">Th25</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">Vx</font> being  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:109_1_2_20"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font>
 <b>and </b><br/><a NAME="E8:109_1_2_20"/><i><font color="Green">E290</font></i>: 
( <font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/><a NAME="E9:109_1_2_20"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">Vx</font> is <a href="rltopsp1.html#V2" target="_self">symmetric</a>
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a></i>;<br/><a NAME="E10:109_1_2_20"/><i><font color="Green">E292</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_2_20_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:109_1_2_20_2"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:109_1_2_20_2"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="txt" href="rltopsp1.html#E40"><i><font color="Green">Lemma103</font></i></a></i>;<br/><a NAME="E6:109_1_2_20_2"/>
<font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:109_1_2_20_2"/><i><font color="Green">E295</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E9:109_1_2_20_2"/><i><font color="Green">E296</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E10:109_1_2_20_2"/><i><font color="Green">E297</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T26" target="_self">Th26</a>, <a class="txt" href="rltopsp1.html#E40"><i><font color="Green">Lemma103</font></i></a></i>;<br/><a NAME="E11:109_1_2_20_2"/>
<font color="Maroon">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FVx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E37"><i><font color="Green">Lemma100</font></i></a></i>;<br/><a NAME="E12:109_1_2_20_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E43"><i><font color="Green">Lemma107</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E13:109_1_2_20_2"/><b>then </b>
 <a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T27" target="_self">Th27</a>, </i>;<br/><a NAME="E14:109_1_2_20_2"/><b>then </b><i><font color="Green">E298</font></i>: 
<font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Maroon">Vx</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D6" target="_self">Def6</a>, , </i>;<br/><font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span> = 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">v</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rltopsp1.html#T29" target="_self">Th29</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E16:109_1_2_20_2"/>
contradiction
 <b>by </b><i><a class="ref" href="rltopsp1.html#T28" target="_self">Th28</a>, <a class="txt" href="rltopsp1.html#E42"><i><font color="Green">Lemma106</font></i></a>, <a class="ref" href="rltopsp1.html#D7" target="_self">Def7</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E12:109_1_2_20"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E13:109_1_2_20"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D5" target="_self">Def5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E45:109_1_2"/><b>then </b>
 <a href="setfam_1.html#K3">INTERSECTION</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/><a NAME="E46:109_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/><a NAME="E47:109_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="setfam_1.html#T36">SETFAM_1:36</a></i>;<br/><a NAME="E48:109_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font></span>)</span> <a href="rusub_4.html#K7">+</a> <font color="Olive">Vx</font></span>)</span> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">Vx</font> is  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> : ( <font color="Olive">x</font> <a href="rusub_4.html#K5">+</a> <font color="Olive">Vx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">Vx</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E49:109_1_2"/>
<font color="Maroon">K</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
