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<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<a NAME="E1:8"/><i><font color="Green">E25</font></i>: 
 <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K8">Filt</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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="E1:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:8_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E5:8_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">x1</font> is  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T28">YELLOW_7:28</a></i>;<br/>
<a NAME="E6:8_1"/>
<font color="Maroon">x1</font> is  <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T26">YELLOW_7:26</a></i>;<br/>
<a NAME="E7:8_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> : verum </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#D24">WAYBEL_0:def 24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:8"/>
 <a href="waybel_0.html#K8">Filt</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><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="E1:8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#D24">WAYBEL_0:def 24</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E4:8_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E5:8_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x1</font> is  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T28">YELLOW_7:28</a></i>;<br/>
<a NAME="E6:8_2"/>
<font color="Maroon">x1</font> is  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T26">YELLOW_7:26</a></i>;<br/>
<a NAME="E7:8_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : verum </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
 <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K8">Filt</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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