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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_1.html#V2" title="WAYBEL_1:attr.2">distributive</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">l</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> holds <br/>( <font color="Olive" title="b1">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> iff  ex <font color="Olive" title="b2">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Olive" title="b1">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b2">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> implies ( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> iff  ex <font color="Olive" title="b1">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  ) )</span><br/>



<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> iff  ex <font color="Olive" title="b1">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<b>set </b><font color="Maroon" title="c3">F</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<b>thus </b><a NAME="E2:49"/>
( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> implies  ex <font color="Olive" title="b1">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<a NAME="E2:49_1"/><i><font color="Green" title="E32">A3</font></i>: 
( ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( not  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="waybel_3.html#K1" title="WAYBEL_3:func.1">waybelow</a> <font color="Olive" title="b1">x</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V2" title="WAYBEL_3:attr.2">satisfying_axiom_of_approximation</a> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> ) )</span><br/><b>assume </b><a NAME="E1:49_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> )</span><br/><a NAME="E2:49_1_1"/><b>then </b>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:49_1_1"/><b>then </b>
not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:49_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> )
 <b>by </b><i><a class="txt" href="#E2:49_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="waybel_3.html#T24" title="WAYBEL_3:th.24">WAYBEL_3:24</a></i>;<br/><a NAME="E6:49_1_1"/>
not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="#E5:49_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/><a NAME="E7:49_1_1"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E8:49_1_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="#E5:49_1_1"><i><font color="Green" title="E33">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">F</font> = <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c2">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E1:49_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_6.html#D1" target="_self" title="WAYBEL_6:def.1">Def1</a>, <a class="ref" href="waybel_6.html#T26" target="_self" title="WAYBEL_6:th.26">Th26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<a NAME="E5:49_1"/>
<font color="Maroon" title="c2">l</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font>
 
;<br/>
<a NAME="E6:49_1"/><b>then </b><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:49_1_2"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="orders_2.html#NR4" title="ORDERS_2:NR.4">&gt;</a> <font color="Maroon" title="c2">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E3:49_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="#E2:49_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:49_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:49_1"/>
<font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Maroon" title="c4">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E6:49_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="waybel_4.html#T56" title="WAYBEL_4:th.56">WAYBEL_4:56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:49"/>
(  ex <font color="Olive" title="b1">F</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">F</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2"/>
 ex <font color="Olive" title="b1">O</font> being <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> st <font color="Maroon" title="c2">l</font> <a href="waybel_4.html#R3" title="WAYBEL_4:pred.3">is_maximal_in</a> <font color="Olive" title="b1">O</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a></span><br/>

<a NAME="E2:49_2"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V2" title="WAYBEL_6:attr.2">meet-irreducible</a>
 
<b>by </b><i><a class="ref" href="waybel_6.html#T13" target="_self" title="WAYBEL_6:th.13">Th13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> implies <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> )</span><br/><b>assume </b><a NAME="E1:49_2_1"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> implies <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> )</span><br/><b>assume </b><a NAME="E2:49_2_1"/><i><font color="Green" title="E33">A9</font></i>: 
( not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><font color="Maroon" title="c2">l</font> = 
<font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:49_2_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">y</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#T6" title="WAYBEL_1:th.6">WAYBEL_1:6</a></i>
;<br/><a NAME="E4:49_2_1"/><b>then </b>
( <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">l</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="txt" href="#E2:49_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="waybel_6.html#D2" target="_self" title="WAYBEL_6:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E5:49_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:49_2_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49_2"/>
<font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#D6" target="_self" title="WAYBEL_6:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
