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

<b>let </b><font color="Maroon">L</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>thus </b><a NAME="E2:27"/>
 <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">Y</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="27_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:27_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:27_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<a NAME="E5:27_1"/>
( <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>


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

<b>thus </b><a NAME="E3:27"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:27_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E4:27_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<a NAME="E5:27_2"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:27_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>


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

<b>thus </b><a NAME="E4:27"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Y</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="27_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:27_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:27_3"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<a NAME="E5:27_3"/>
( <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  &amp; <font color="Maroon">y</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<b>hence </b><a NAME="E6:27_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E5:27"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">Y</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span><b> such that </b><br/><a NAME="E8:27"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/>
<a NAME="E9:27"/>
(  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="lattice3.html#K9">~</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:27"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>


</div>
