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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_1.html#V2">distributive</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:48"/>
( <font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font> implies for <font color="Olive">P</font> being <a href="waybel_7.html#V1" target="_self">prime</a> <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">P</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T20">WAYBEL_3:20</a></i>;<br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">P</font> being <a href="waybel_7.html#V1" target="_self">prime</a> <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">P</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">P</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">P</font> is <a href="waybel_7.html#V1" target="_self">prime</a> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/><a NAME="E4:48_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E5:48_1"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T4" target="_self">Th4</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:48_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T21">WAYBEL_3:21</a></i>;<br/>


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