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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V2">compact</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<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">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">X</font> );<br/>
<a NAME="E1:84"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">u</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E2:84_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:84_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:84_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 ;<br/>
<b>thus </b><a NAME="E4:84_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> ;<br/>
<b>take </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
;<br/>

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

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

<b>thus </b><a NAME="E4:84_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:84"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E4:84"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">RF</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E6:84"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">RF</font> is <a href="compts_1.html#V1">centered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_2"/>
<font color="Maroon">RF</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="relat_1.html#T65">RELAT_1:65</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D2">COMPTS_1:def 2</a><br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:84_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:84_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">RF</font>
 <b>and </b><br/><a NAME="E4:84_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">H</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">H1</font> = <font color="Maroon">H</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> :  ex <font color="Olive">h</font>, <font color="Olive">Ch</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Olive">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">h</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H1</font>;<br/>
<a NAME="E7:84_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RF</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:84_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E10:84_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="ref" href="waybel_9.html#T3" target="_self">Th3</a></i>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E13:84_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E14:84_2"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<a NAME="E15:84_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> :  ex <font color="Olive">h</font>, <font color="Olive">Ch</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Olive">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">h</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> :  ex <font color="Olive">h</font>, <font color="Olive">Ch</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Olive">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">h</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>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">h</font>, <font color="Olive">Ch</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">Ch</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Olive">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">h</font> );<br/>
<a NAME="E17:84_2"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H1</font>  ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H1</font>;<br/>

<a NAME="E1:84_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RF</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_2_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:84_2_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:84_2_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:84_2_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span> 
 <b>and </b><br/><a NAME="E9:84_2_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:84_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">J</font> ;<br/>
<b>take </b><font color="Maroon">u</font> = <font color="Maroon">a</font>;<br/>

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

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

<b>take </b><font color="Maroon">Ch</font> =  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>;<br/>

<b>thus </b><a NAME="E12:84_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:84_2_1"/>
<font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E14:84_2_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E15:84_2_1"/>
<font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">h</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Q</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">H1</font>,<font color="Maroon">J</font><b> such that </b><br/><a NAME="E19:84_2"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H1</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">e</font>,<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E20:84_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:84_2"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Q</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T2" target="_self">Th2</a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E22:84_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:84_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_2_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E4:84_2_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">H1</font> <b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Ch</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:84_2_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:84_2_2"/>
( <font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  &amp; <font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">h</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:84_2_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">RQ</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Q</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font></span>)</span> ;<br/>
<a NAME="E24:84_2"/>
( not  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">i0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E26:84_2"/>
<font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E27:84_2"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">i0</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">RQ</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<a NAME="E28:84_2"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:84_2_3"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:84_2_3"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">Ch</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:84_2_3"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:84_2_3"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:84_2_3"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span> 
 <b>and </b><br/><a NAME="E7:84_2_3"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">h</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:84_2_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:84_2_3"/><b>then </b>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">i0</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E10:84_2_3"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:84_2_3"/>
<font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E12:84_2_3"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T4" target="_self">Th4</a>, <a class="txt" href="waybel_9.html#E7"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="waybel_9.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E29:84_2"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">H</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="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:84"/>
<font color="Maroon">RF</font> is <a href="tops_2.html#V2">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><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">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E1:84_3"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RF</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_3"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:84_3"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:84_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span>  )
 <b>and </b><br/><a NAME="E8:84_3"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:84_3"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="waybel_9.html#T2" target="_self">Th2</a>, <a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<b>hence </b><a NAME="E10:84_3"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:84"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RF</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:84"><i><font color="Green">E40</font></i></a>, <a class="ref" href="compts_1.html#T13">COMPTS_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:84"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">RF</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>assume </b><a NAME="E12:84"/>
not <font color="Maroon">c</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c</font><b> such that </b><br/><a NAME="E14:84"/><i><font color="Green">E156</font></i>: 
not <font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E3:84"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E15:84"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T10">WAYBEL_0:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">s0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E17:84"/><i><font color="Green">E157</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">s0</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E19:84"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font>
 <b>and </b><br/><a NAME="E20:84"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> </span>}</span> 
 <b>and </b><br/><a NAME="E21:84"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s0</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>
<a NAME="E22:84"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">RF</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E23:84"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E24:84"/><i><font color="Green">E162</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E25:84"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_4"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">q</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="E2:84_4"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E3:84_4"/>
not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:84_4"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:84_4"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E7:84_4"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">s0</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E8:84_4"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E9:84_4"/><b>then </b>
not <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E10:84_4"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:84"/><b>then </b>
<font color="Maroon">O</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E27:84"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E28:84"/><i><font color="Green">E167</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<a NAME="E29:84"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</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="E30:84"/>
contradiction
 <b>by </b><i>, <a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>


</div>
