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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:91"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V15" target="_self">principal</a> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6" target="_self">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:91_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>2</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D22" target="_self">WAYBEL_0:def 22</a><br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E2:91_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6" target="_self">uparrow</a> <font color="Maroon">c<sub>3</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:91_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:91_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E1:91_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_0.html#E1:91_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:91_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6" target="_self">uparrow</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18" target="_self">Th18</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></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="E3:91_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6" target="_self">uparrow</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E5:91_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E3:91_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E6:91_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E1:91_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#D20" target="_self">Def20</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:91"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6" target="_self">uparrow</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D22" target="_self">WAYBEL_0:def 22</a><br/>

<a NAME="E3:91"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:91"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E2:91"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#T18" target="_self">Th18</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>thus </b><a NAME="E5:91"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E2:91"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_0.html#T18" target="_self">Th18</a></i>;<br/>


</div>
