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

<b>let </b><font color="Maroon">T</font> be  <a href="waybel_3.html#V3">continuous</a> <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="yellow_0.html#V5">meet-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E26</font></i>: 
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>given </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E2:59"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E4:59"/>
<font color="Maroon">S</font> is <a href="waybel_0.html#V3">filtered-infs-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be  <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D3">WAYBEL_0:def 3</a><br/>

<b>assume </b><a NAME="E1:59_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/>
<b>set </b><font color="Maroon">N</font> = <font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> ;<br/>
<b>assume </b><a NAME="E3:59_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">T</font>
 ;<br/>

<a NAME="E4:59_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="waybel19.html#T27">WAYBEL19:27</a></i>;<br/>
<a NAME="E5:59_1"/><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">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<a NAME="E6:59_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a>  <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E6:59_1"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E7:59_1"/>
 <a href="yellow_6.html#K11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <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"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="waybel19.html#T43">WAYBEL19:43</a></i>;<br/>
<a NAME="E8:59_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="waybel19.html#T26">WAYBEL19:26</a></i>;<br/>
<a NAME="E9:59_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">F</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E10:59_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E6:59_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:59"/>
<font color="Maroon">S</font> is <a href="yellow_0.html#V7">infs-inheriting</a>
 <b>by </b><i>, </i>;<br/>


</div>
