<?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>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/>
<b>reconsider </b><font color="Maroon">C</font> =  <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> as  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a>  <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="E2:81"/>
<font color="Maroon">C</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D4">WAYBEL_0:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:81_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</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/>

<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> 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 <font color="Maroon">C</font> <b>by </b><i><a class="ref" href="waybel25.html#T2" target="_self">Th2</a></i>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font>;<br/>
<a NAME="E4:81_1"/><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/>
<a NAME="E5:81_1"/><i><font color="Green">E44</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/>
<a NAME="E6:81_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="waybel25.html#K2" target="_self">commute</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>,<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>
 
<b>by </b><i/>;<br/>
<a NAME="E7:81_1"/><b>then </b>
 <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 <span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span></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/>;<br/>
<a NAME="E8:81_1"/><b>then </b>
 <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 <span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span></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>,<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="txt" href="waybel25.html#E9:81_1"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>
<a NAME="E9:81_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <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 <span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span></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> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="waybel24.html#D3">WAYBEL24:def 3</a></i>;<br/>
<a NAME="E10:81_1"/>
 <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#G1">NetStr</a>(# <font color="Maroon">D</font>,<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">C</font> <a href="toler_1.html#K1">|_2</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font></span>)</span> <a href="nattra_1.html#K1">|</a> <font color="Maroon">D</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="waybel17.html#D4">WAYBEL17:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:81_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">D</font>,<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> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="waybel25.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow14.html#T2">YELLOW14:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:81"/>
 <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="waybel_0.html#V4">directed-sups-inheriting</a>  <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>
 ;<br/>


</div>
