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<b>set </b><font color="Maroon">F</font> =  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>;<br/>
<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K3">BoolePoset</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span>;<br/>
<i><font color="Green">E31</font></i>:  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> = 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>
<br/>.= 
 <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span></span>)</span> #)
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>
;<br/>
<a NAME="E2:15_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:15_1_1_2"/>
 <a href="xboole_0.html#K1">{}</a>  <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#T2">XBOOLE_1:2</a></i>;<br/><b>assume </b><a NAME="E2:15_1_1_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>
 ;<br/><a NAME="E3:15_1_1_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>then 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="E5:15_1_1_2"/><i><font color="Green">E37</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">i</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>  <a href="xboole_0.html#K1">{}</a> 
 <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">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:15_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</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#D5">YELLOW_6:def 5</a></i>;<br/><b>thus </b><a NAME="E8:15_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:15_1_1"/><b>then </b>
 <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
;<br/>
<b>hence </b><a NAME="E5:15_1_1"/>
 <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i>, , <a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E6:15_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font> )
 ;<br/>

<a NAME="E7:15_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">b</font> )
 
<b>by </b><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="E9:15_1_1"/><i><font color="Green">E55</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">a</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">i2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E11:15_1_1"/><i><font color="Green">E57</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">b</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="E13:15_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">i1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>;<br/>

<a NAME="E14:15_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3"><b>now </b></a><div class="add"><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="E1:15_1_1_3"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:15_1_1_3"/><b>then </b>
( <font color="Maroon">i1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <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="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E3:15_1_1_3"/><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">a</font> &amp; <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">b</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:15_1_1_3"/>
<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">a</font> <a href="xboole_0.html#K3">/\</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/></div><b>end;</b></div>
<a NAME="E16:15_1_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>hence </b><a NAME="E17:15_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/>

<a NAME="E18:15_1_1"/>
( <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E19:15_1_1"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>


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