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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon" title="c2">x</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c3">I</font> = <font color="Maroon" title="c2">x</font> as  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> ;<br/><b>reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">I</font> as   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> <b>by </b><i><a class="ref" href="yellow_7.html#T26" title="YELLOW_7:th.26">YELLOW_7:26</a>, <a class="ref" href="yellow_7.html#T28" title="YELLOW_7:th.28">YELLOW_7:28</a></i>;<br/><a NAME="E4:26_1"/>
<font color="Maroon" title="c4">F</font> is <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D2" target="_self" title="WAYBEL_7:def.2">WAYBEL_7:def 2</a><br/>

<a NAME="E1:26_1_1"/>
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a>, <a class="ref" href="yellow_7.html#T22" title="YELLOW_7:th.22">YELLOW_7:22</a></i>;<br/>
<b>hence </b><a NAME="E2:26_1_1"/>
(  not <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#D1" target="_self" title="WAYBEL_7:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:26_1"/>
<font color="Maroon" title="c2">x</font> is  <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:26"/>
<font color="Maroon" title="c2">x</font> is  <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">I</font> = <font color="Maroon" title="c2">x</font> as  <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">I</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_7.html#T26" title="YELLOW_7:th.26">YELLOW_7:26</a>, <a class="ref" href="yellow_7.html#T28" title="YELLOW_7:th.28">YELLOW_7:28</a></i>;<br/>
<a NAME="E5:26"/>
<font color="Maroon" title="c4">F</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D1" target="_self" title="WAYBEL_7:def.1">WAYBEL_7:def 1</a><br/>

<a NAME="E1:26_2"/>
( <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c6">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">y</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a>, <a class="ref" href="yellow_7.html#T21" title="YELLOW_7:th.21">YELLOW_7:21</a></i>;<br/>
<b>hence </b><a NAME="E2:26_2"/>
(  not <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#D2" target="_self" title="WAYBEL_7:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:26"/>
<font color="Maroon" title="c2">x</font> is  <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>
 ;<br/>


</div>
