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

<b>let </b><font color="Maroon">L</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><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>set </b><font color="Maroon">P</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span>;<br/>
<a NAME="E1:29"/><i><font color="Green">E33</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E2:29"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">I</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:29"/>
( <font color="Maroon">I</font> is <a href="waybel_7.html#V1" target="_self">prime</a> implies <font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  holds <br/>(  not <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D1" target="_self">WAYBEL_7:def 1</a><br/>

<a NAME="E2:29_1"/>
<font color="Maroon">i</font> is <a href="waybel_6.html#V5">prime</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D6">WAYBEL_6:def 6</a><br/>

<a NAME="E1:29_1_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> &amp; <font color="Maroon">y</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/>;<br/>
<a NAME="E2:29_1_1"/><b>then </b>
(  ex <font color="Olive">J</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp;  ex <font color="Olive">J</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font>, <font color="Maroon">Y</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> ;<br/>
<b>assume </b><a NAME="E4:29_1_1"/>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E5:29_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E6:29_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T45">YELLOW_2:45</a></i>;<br/>
<b>assume </b><a NAME="E7:29_1_1"/>
( not <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font> &amp; not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E8:29_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( not <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> &amp; not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</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="E10:29_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:29_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, </i>;<br/>
<a NAME="E14:29_1_1"/>
( <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E15:29_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>
<a NAME="E16:29_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E17:29_1_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_6.html#D7">WAYBEL_6:def 7</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:29"/>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a>  <a href="waybel_6.html#K4">PRIME</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E6:29"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> is <a href="waybel_6.html#V5">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_6.html#D7">WAYBEL_6:def 7</a></i>;<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D1" target="_self">WAYBEL_7:def 1</a><br/>

<b>reconsider </b><font color="Maroon">X</font> =  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>, <font color="Maroon">Y</font> =  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E8:29"/>
( <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> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font>, <font color="Maroon">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E10:29"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 
;<br/>
<a NAME="E11:29"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E12:29"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T37">YELLOW_4:37</a></i>;<br/>
<a NAME="E13:29"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="yellow_2.html#T45">YELLOW_2:45</a>, <a class="ref" href="yellow_4.html#T50">YELLOW_4:50</a></i>;<br/>
<b>assume </b><a NAME="E14:29"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E15:29"/>
<font color="Maroon">X</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:29_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:29_2"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then 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> ;<br/>
<a NAME="E4:29_2"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>
<a NAME="E5:29_2"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E6:29_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:29"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">Y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E17:29"/><b>then </b>
( <font color="Maroon">X</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font> or <font color="Maroon">Y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_6.html#D6">WAYBEL_6:def 6</a></i>;<br/>
<a NAME="E18:29"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> or <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<b>hence </b><a NAME="E19:29"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> or <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/>


</div>
