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

<b>set </b><font color="Maroon">IT</font> =  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font>;<br/>
<a NAME="E1:113_1_1"/><i><font color="Green">E29</font></i>: 
the <a href="pre_topc.html#U1">topology</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> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> : for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</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> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V</font> =  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:113_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E4:113_1_1"/><b>then </b>
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</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> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:113_1_1"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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/>;<br/>
<b>hence </b><a NAME="E6:113_1_1"/>
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#R2">in</a> the <a href="pre_topc.html#U1">topology</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM3">Subset-Family</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="E1:113_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</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>reconsider </b><font color="Maroon">V</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1_1"><b>now </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">S</font>;<br/><b>assume </b><a NAME="E1:113_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</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="E3:113_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E4:113_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<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="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E5:113_1_1_1_1"/><i><font color="Green">E55</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>
 ;<br/><a NAME="E6:113_1_1_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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>, </i>;<br/><b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:113_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E9:113_1_1_1_1"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</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> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/><a NAME="E10:113_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_6.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E11:113_1_1_1_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E12:113_1_1_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</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="waybel_0.html#T8">WAYBEL_0:8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:113_1_1_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</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>, <font color="Maroon">b</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>assume </b><a NAME="E8:113_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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>then consider </b><font color="Maroon">Va</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E10:113_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Va</font>
 <b>and </b><br/><a NAME="E11:113_1_1"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Va</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</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> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">Va</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E12:113_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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>then consider </b><font color="Maroon">Vb</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E14:113_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Vb</font>
 <b>and </b><br/><a NAME="E15:113_1_1"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Vb</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</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> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">Vb</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_2"><b>now </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">S</font>;<br/><b>assume </b><a NAME="E1:113_1_1_2"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E2:113_1_1_2"/><i><font color="Green">E149</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>
 ;<br/><a NAME="E3:113_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:113_1_1_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">Va</font>
 <b>by </b><i>, , <a class="ref" href="yellow_6.html#T12" target="_self">Th12</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:113_1_1_2"/><i><font color="Green">E181</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">i1</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> <font color="Maroon">Va</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/><a NAME="E7:113_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:113_1_1_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">Vb</font>
 <b>by </b><i>, , <a class="ref" href="yellow_6.html#T12" target="_self">Th12</a></i>;<br/><b>then consider </b><font color="Maroon">i2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E10:113_1_1_2"/><i><font color="Green">E182</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">i2</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> <font color="Maroon">Vb</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">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E12:113_1_1_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">i1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E13:113_1_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">i2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E14:113_1_1_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">i</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

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

<b>assume </b><a NAME="E1:113_1_1_2_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:113_1_1_2_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E3:113_1_1_2_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<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">Va</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</font></i></a></i>;<br/>
<a NAME="E4:113_1_1_2_1"/>
<font color="Maroon">i2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#D6" target="_self">Def6</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E5:113_1_1_2_1"/><b>then </b>
<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">Vb</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:113_1_1_2_1"/>
<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">V</font>
 <b>by </b><i>, , <a class="ref" href="yellow_6.html#D7" target="_self">Def7</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:113_1_1"/>
<font color="Maroon">a</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</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/>;<br/>


</div>
