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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V1">lower-bounded</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">F</font> is <a href="tex_2.html#V1">proper</a>
 ;<br/><a NAME="E2:6_1"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:6_1"/><i><font color="Green">E33</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) )
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E6:6_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/><b>hence </b><a NAME="E7:6_1"/>
not  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:6"/>
not  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:6"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon">F</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/>


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