<?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="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="yellow_6.html#V8" target="_self">topological</a>  <a href="yellow_6.html#M4" target="_self">Convergence-Class</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:120"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">S</font>
 ;<br/>

<b>set </b><font color="Maroon">CC</font> =  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span>] <b>means </b> ex <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> );<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] </span>}</span> ;<br/>
<a NAME="E2:120"/>
<span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E4:120"/>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><br/> for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<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>assume </b><b>that </b><br/><a NAME="E1:120_1"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:120_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:120_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:120_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T12" target="_self">Th12</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">N</font> <a href="yellow_6.html#K6" target="_self">"</a> <font color="Maroon">F</font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font> <b>by </b><i><a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</font></i></a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  ex <font color="Olive">X</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> ) );<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/><a NAME="E1:120_1_1"/><i><font color="Green">E56</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><a NAME="E2:120_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 ;<br/><a NAME="E3:120_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E4:120_1_1"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><a NAME="E5:120_1_1"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E6:120_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E8:120_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E9:120_1_1"/><i><font color="Green">E118</font></i>: 
 ex <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> )
 ;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font><b> such that </b><br/><a NAME="E11:120_1_1"/><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E12:120_1_1"/><i><font color="Green">E127</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E13:120_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">N</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E15:120_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:120_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_6.html#T13" target="_self">Th13</a></i>;<br/>

<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">N</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E3:120_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<b>thus </b><a NAME="E4:120_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">J</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E8:120_1"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1">MSUALG_9:sch 1</a>();<br/></i>
<a NAME="E9:120_1"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> holds <br/><font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:120_1_2"/><b>then </b>
 ex <font color="Olive">X</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:120_1_2"/>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as    <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,<font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E11:120_1"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_3"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:120_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>

<a NAME="E2:120_1_3"/>
 ex <font color="Olive">X</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:120_1_3"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">I</font> =  <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E13:120_1"/><i><font color="Green">E147</font></i>: 
 ex <font color="Olive">N0</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">N0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N0</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:120_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:120_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</a>, <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/>
<a NAME="E16:120_1"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:120_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T12" target="_self">Th12</a>, <a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a></i>;<br/>
<a NAME="E18:120_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a>, </i>;<br/>
<b>set </b><font color="Maroon">XX</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : verum </span>}</span> ;<br/>
<a NAME="E19:120_1"/><i><font color="Green">E181</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">I</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"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E20:120_1"/>
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> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : verum </span>}</span>  holds <br/><font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_4"><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="E1:120_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:120_1_4"/><b>then </b>
 ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E3:120_1_4"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:120_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>
<a NAME="E22:120_1"/><b>then </b><i><font color="Green">E182</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E24:120_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E25:120_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>hence </b><a NAME="E26:120_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T13" target="_self">Th13</a>, <a class="ref" href="yellow_6.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:120"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">F</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:120"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2"><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="E1:120_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<a NAME="E3:120_2"/>
 <a href="xboole_0.html#K1">{}</a>  <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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="relset_1.html#NM3">Relation</a> of <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="relset_1.html#T8">RELSET_1:8</a></i>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #);<br/>
<div><i><font color="Green">E185</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #);<br/><a NAME="E1:120_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y</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/><a NAME="E2:120_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:120_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:120_2"/><i><font color="Green">E186</font></i>: 
 <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #) is <a href="orders_2.html#V3">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #), <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>thus </b><a NAME="E1:120_2_2"/>
(  not <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> or  not <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> or <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:120_2"/>
 <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #) is <a href="waybel_0.html#V7">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5" target="_self">YELLOW_6:def 5</a><br/>

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

<b>thus </b><a NAME="E1:120_2_3"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> =  <a href="orders_2.html#G1">RelStr</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>,<font color="Maroon">r</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a> <a href="waybel_0.html#V7">directed</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">N</font> = <font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font>;<br/>
<a NAME="E9:120_2"/><i><font color="Green">E187</font></i>: 
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:120_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font> as  <a href="waybel_0.html#V6">strict</a> <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E12:120_2"/>
 <a href="yellow_6.html#K1" target="_self">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:120_2"/><b>then </b><i><font color="Green">E190</font></i>: 
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E14:120_2"/><i><font color="Green">E191</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #)
 
<b>by </b><i/>;<br/>
<a NAME="E15:120_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="classes2.html#T62">CLASSES2:62</a></i>;<br/>
<a NAME="E16:120_2"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="classes2.html#T3">CLASSES2:3</a></i>;<br/>
<a NAME="E17:120_2"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</a>, </i>;<br/>
<a NAME="E18:120_2"/><b>then </b><i><font color="Green">E193</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a>, </i>;<br/>
<a NAME="E19:120_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:120_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">q</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E21:120_2"/><b>then </b><i><font color="Green">E195</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E22:120_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E23:120_2"/>
<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="yellow_6.html#T13" target="_self">Th13</a>, <a class="ref" href="yellow_6.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:120"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T12" target="_self">Th12</a>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<a NAME="E8:120"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:120"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp;  ex <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> ) )
 
;<br/>
<b>hence </b><a NAME="E10:120"/>
 ex <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Olive">N</font> )
 ;<br/>


</div>
