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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  <a href="waybel25.html#V1" target="_self">monotone-convergence</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of  <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:80"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds   <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Olive">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 ;<br/>

<b>set </b><font color="Maroon">m</font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E2:80"/><i><font color="Green">E43</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">Y</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> #)
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">fo</font> =  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19">WAYBEL24:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">fo</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E5:80"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</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/>
<a NAME="E6:80"/>
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>  st <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> ;<br/>
<a NAME="E2:80_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><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>thus </b><a NAME="E1:80_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:80_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">NET</font> =  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/>
<a NAME="E4:80_1_1_1"/><i><font color="Green">E64</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:80_1_1_1"/>
 <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> is    <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<a NAME="E6:80_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E7:80_1_1_1"/><b>then </b><i><font color="Green">E68</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"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></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">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #)
 
<b>by </b><i/>;<br/>
<a NAME="E8:80_1_1_1"/>
 <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 
<b>by </b><i><a class="ref" href="waybel25.html#T2" target="_self">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_2.html#T18">WAYBEL_2:18</a></i>;<br/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel25.html#T2" target="_self">Th2</a>, <a class="txt" href="waybel25.html#E5:80"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">W</font>

;<br/>
<a NAME="E11:80_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="waybel25.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:80_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E14:80_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:80_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E17:80_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E2:80_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="txt" href="waybel25.html#E1:80_1_1_1"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T21">WAYBEL24:21</a></i>;<br/>
<a NAME="E21:80_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:80_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:80_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="waybel25.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E24:80_1_1_1"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E25:80_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:80_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:80_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E5:80_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:80_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E8:80_1_1"/><i><font color="Green">E93</font></i>: 
 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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> )
 <b>by </b><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">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E10:80_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E11:80_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<a NAME="E12:80_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E13:80_1_1"/><i><font color="Green">E117</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">X</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">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<a NAME="E15:80_1_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds   <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Olive">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:80_1_1_2"/>
 <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> is <a href="waybel_0.html#V10">eventually-directed</a>
 
<b>by </b><i><a class="ref" href="waybel25.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E2:80_1_1_2"/>
 <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:80_1_1"/><b>then </b><i><font color="Green">E118</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:80_1_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:80_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
 
;<br/>
<a NAME="E19:80_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">fo</font>
 
<b>by </b><i>, , <a class="txt" href="waybel25.html#E2:80"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E20:80_1_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fo</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E22:80_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E23:80_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">O</font>
 <b>by </b><i/>;<br/>
<a NAME="E24:80_1_1"/>
<font color="Maroon">V</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="waybel25.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel25.html#T4" target="_self">Th4</a>, , <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:80_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<a NAME="E27:80_1_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E28:80_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E29:80_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:80_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E4:80_1"/>
<span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</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="E1:80_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:80_1_2"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp;  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:80_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> :  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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E7:80_1"/>
<font color="Maroon">M</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_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">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:80_1_3"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:80_1_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:80_1_3"/><i><font color="Green">E126</font></i>: 
 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">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> )
 ;<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">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E6:80_1_3"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E7:80_1_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:80_1_3"/><i><font color="Green">E129</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #)
 
;<br/>
<b>consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E10:80_1_3"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">g'</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g''</font> = <font color="Maroon">g'</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="waybel25.html#E1:80_1"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="waybel25.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>
<a NAME="E12:80_1_3"/>
<font color="Maroon">g''</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel25.html#T3" target="_self">Th3</a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E13:80_1_3"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="waybel25.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:80_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:80"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
