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<b>let </b><font color="Maroon">T</font> be  <a href="yellow_0.html#V2">upper-bounded</a> <a href="waybel_0.html#NM6">Semilattice</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="E1:20"/><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> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V3">filtered-infs-inheriting</a> )
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <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="yellow_0.html#D18">YELLOW_0:def 18</a><br/>

<a NAME="E2:20"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="tarski.html#R1">c=</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_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font>;<br/>
<b>assume </b><a NAME="E4:20"/><i><font color="Green">E30</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">A</font>,<font color="Maroon">T</font>
 ;<br/>

<a NAME="E5:20"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T60">WAYBEL_0:60</a></i>;<br/>
<a NAME="E6:20"/>
 <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><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="E1:20_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">T</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>assume </b><a NAME="E6:20_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<a NAME="E7:20_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:20_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> =  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</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">S</font> <b>by </b><i><a class="ref" href="waybel10.html#T24">WAYBEL10:24</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font>;<br/><a NAME="E1:20_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E2:20_2"/>
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T55">YELLOW_0:55</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><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 <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:20_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:20_3"/><b>then </b>
 ex <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> )
 ;<br/><b>hence </b><a NAME="E3:20_3"/>
 ex <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font> st <br/>(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<font color="Maroon">T</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">C</font>;<br/><b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>assume </b><a NAME="E2:20_4"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:20_4"/><b>then </b>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Z</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T55">YELLOW_0:55</a></i>;<br/><b>hence </b><a NAME="E4:20_4"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E12:20"/><b>then </b>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_0.html#T58">WAYBEL_0:58</a></i>;<br/>
<b>hence </b><a NAME="E13:20"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">A</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="ref" href="waybel_0.html#D3">WAYBEL_0:def 3</a></i>;<br/>


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