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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E23">A1</font></i>: 
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E3:38_1"/>
<font color="Maroon" title="c5">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="waybel_7.html#E1:38_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>;<br/><b>hence </b><a NAME="E4:38_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel_7.html#E1:38"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="waybel_7.html#E1:38_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:38"/><b>then </b>
<font color="Maroon" title="c2">I</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">P</font> being   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:38"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c4">P</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> &amp; <font color="Maroon" title="c4">P</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T27" target="_self" title="WAYBEL_7:th.27">Th27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">P</font>
;<br/>

<b>thus </b><a NAME="E6:38"/>
( <font color="Maroon" title="c4">P</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> &amp; <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E5:38"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>

<b>assume </b><a NAME="E7:38"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 ;<br/>

<a NAME="E8:38"/><b>then </b>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E5:38"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:38"/><b>then </b>
not <font color="Maroon" title="c3">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>;<br/>
<b>hence </b><a NAME="E10:38"/>
contradiction
 ;<br/>


</div>
