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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> ;<br/>
<a NAME="E1:76"/>
<span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><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="E1:76_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:76_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> )
 ;<br/>
<b>thus </b><a NAME="E4:76_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X'</font> = <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:76"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/>
<a NAME="E4:76"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 
;<br/>
<b>thus </b><a NAME="E5:76"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</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="76_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="E1:76_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<a NAME="E2:76_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">X'</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:76_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:76_2_1"/><b>then </b>
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E3:76_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D6" target="_self">Def6</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:76_2"/><b>then </b>
<font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T21">WAYBEL_3:21</a></i>;<br/>
<b>hence </b><a NAME="E6:76_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T7">WAYBEL_3:7</a></i>;<br/>


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

<b>thus </b><a NAME="E6:76"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><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="E1:76_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:76_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_3.html#D3">WAYBEL_3:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:76_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> )
 ;<br/>
<a NAME="E5:76_3"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span>  holds <br/><font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_3_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">I</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:76_3_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font> )
 ;<br/>
<b>thus </b><a NAME="E4:76_3_1"/>
<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="waybel_4.html#D6" target="_self">Def6</a>, <a class="ref" href="waybel_4.html#D7" target="_self">Def7</a>, <a class="ref" href="waybel_3.html#T20">WAYBEL_3:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:76_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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


</div>
