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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V2" title="YELLOW_0:attr.2">upper-bounded</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 <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Olive" title="b1">l</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> )</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 <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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> ) )</span><br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E28">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 <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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> )</span><br/>

<b>set </b><font color="Maroon" title="c3">X1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>thus </b><a NAME="E3:45"/>
( <font color="Maroon" title="c2">l</font> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a> implies <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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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"> ( <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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E29">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"> <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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font></span><br/>

<div><i><font color="Green" title="E30">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies  ex <font color="Olive" title="b1">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> ) )</span><br/><b>assume </b><a NAME="E1:45_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 ; <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">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span><br/><a NAME="E2:45_1_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; not <font color="Maroon" title="c5">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> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; not <font color="Maroon" title="c6">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="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:45_1_1"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
( not <font color="Maroon" title="c5">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="c6">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="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/><a NAME="E4:45_1_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <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="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</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="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_1_1"/>
<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>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</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="E2:45_1_1_1"/><b>then </b>
<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="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>hence </b><a NAME="E3:45_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="#E3:45_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <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><a NAME="E6:45_1_1"/><b>then </b>
<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">X</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/><b>hence </b><a NAME="E7:45_1_1"/>
 ex <font color="Olive" title="b1">z</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">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="#E4:45_1_1"><i><font color="Green" title="E32">A5</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>
<div><i><font color="Green" title="E31">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>now </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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2"/><i><font color="Green" title="E32">A7</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:45_1_2"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp; not <font color="Maroon" title="c5">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="E3:45_1_2"/><b>then </b>
not <font color="Maroon" title="c5">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/><a NAME="E4:45_1_2"/><b>then </b>
not <font color="Maroon" title="c6">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="#E1:45_1_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E5:45_1_2"/><b>then </b>
not <font color="Maroon" title="c6">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="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/><b>hence </b><a NAME="E6:45_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_3"/>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:45_1_3"/><b>then </b>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</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/><a NAME="E3:45_1_3"/><b>then </b>
 <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">l</font>
 <b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E4:45_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a>, <a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</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="E5:45_1"/>
<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>  is   <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="#E2:45_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="#E3:45_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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="E4:45"/>
( <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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <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> )
  <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="45_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_2"/><i><font color="Green" title="E29">A8</font></i>: 
<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>  is   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</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> is <a href="waybel_6.html#V5" title="WAYBEL_6:attr.5">prime</a></span><br/>

<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_6.html#D6" target="_self" title="WAYBEL_6:def.6">WAYBEL_6:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> )</span><br/>

<b>assume </b><a NAME="E2:45_2"/><i><font color="Green" title="E30">A9</font></i>: 
<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="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"> ( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">l</font> )</span><br/>

<a NAME="E3:45_2"/>
<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="E4:45_2"/><b>then </b><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c2">l</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="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="45_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2_1"/>
( not <font color="Maroon" title="c5">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="c6">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/><a NAME="E2:45_2_1"/><b>then </b>
( not <font color="Maroon" title="c5">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> &amp; not <font color="Maroon" title="c6">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="ref" href="waybel_0.html#T17" title="WAYBEL_0:th.17">WAYBEL_0:17</a></i>;<br/><a NAME="E3:45_2_1"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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/><b>then consider </b><font color="Maroon" title="c7">z</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="E5:45_2_1"/><i><font color="Green" title="E32">A11</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="#E1:45_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/><a NAME="E6:45_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="#E5:45_2_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E7:45_2_1"/><b>then </b>
<font color="Maroon" title="c7">z</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="#E2:45_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/><a NAME="E8:45_2_1"/><b>then </b>
<font color="Maroon" title="c2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="#E1:45_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="#E5:45_2_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/><b>hence </b><a NAME="E9:45_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:45_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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="E6:45_2"/>
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">l</font> or <font color="Maroon" title="c6">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"> verum</span><br/>


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


</div>
