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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">I</font> being  <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">I</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> iff ( <font color="Olive" title="b1">I</font> <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> or <font color="Olive" title="b1">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> iff ( <font color="Maroon" title="c2">I</font> <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> or <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )</span><br/>



<b>set </b><font color="Maroon" title="c3">F</font> = <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<b>thus </b><a NAME="E1:28"/>
(  not <font color="Maroon" title="c2">I</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> or <font color="Maroon" title="c2">I</font> <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> or <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">I</font> <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> or <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) implies <font color="Maroon" title="c2">I</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E15">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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 <font color="Olive" title="b1">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D1" target="_self" title="WAYBEL_7:def.1">WAYBEL_7:def 1</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">I</font> <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> or <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<a NAME="E2:28_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:28_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<a NAME="E2:28_1_1"/><b>then </b>
( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</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="E3:28_1_1"/><b>then </b>
not <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 
<b>by </b><i><a class="txt" href="#E1:28_1"><i><font color="Green" title="E15">A1</font></i></a></i>;<br/>
<a NAME="E4:28_1_1"/><b>then </b>
( <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <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="c4">x</font> &amp; <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="c5">y</font> )
 
<b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a>, <a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E5:28_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</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>
<a NAME="E3:28_1"/>
<font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><b>proof </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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )</span><br/>

<b>assume </b><a NAME="E1:28_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</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#NR2" title="ORDERS_2:NR.2">&gt;=</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"> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span><br/>

<a NAME="E2:28_1_2"/><b>then </b>
( not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> ) )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</a>, <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="E3:28_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</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="E4:28_1"/>
( <font color="Maroon" title="c2">I</font> <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> or <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="#E2:28_1"><i><font color="Green" title="E16">A2</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>assume </b><a NAME="E2:28"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c2">I</font> <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> or <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">I</font> is <a href="waybel_7.html#V1" title="WAYBEL_7:attr.1">prime</a></span><br/>

<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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D1" target="_self" title="WAYBEL_7:def.1">WAYBEL_7:def 1</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="c4">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )</span><br/>

<b>assume </b><a NAME="E3:28"/>
<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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</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> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )</span><br/>

<a NAME="E4:28"/><b>then </b>
not <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <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/>
<a NAME="E5:28"/><b>then </b>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="waybel_0.html#T41" title="WAYBEL_0:th.41">WAYBEL_0:41</a></i>;<br/>
<b>hence </b><a NAME="E6:28"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <b>by </b><i><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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