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

<b>set </b><font color="Maroon">A</font> =  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>;<br/>
<b>take </b>
 <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E1:7_1_1"/>
 <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font> is <a href="waybel_4.html#V1" target="_self">auxiliary(i)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D8" target="_self">WAYBEL_4:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D4" target="_self">WAYBEL_4:def 4</a><br/>

<b>assume </b><a NAME="E1:7_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E2:7_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E3:7_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E2:7_1_1"/>
 <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font> is <a href="waybel_4.html#V2" target="_self">auxiliary(ii)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D5" target="_self">WAYBEL_4:def 5</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">u</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:7_1_1_2"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E3:7_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 ;<br/>

<a NAME="E4:7_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:7_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E6:7_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E7:7_1_1_2"/><b>then </b>
<font color="Maroon">u</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E8:7_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:7_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 ;<br/>


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

<b>thus </b><a NAME="E3:7_1_1"/>
 <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font> is <a href="waybel_4.html#V3" target="_self">auxiliary(iii)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D6" target="_self">WAYBEL_4:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7_1_1_3"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:7_1_1_3"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E3:7_1_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:7_1_1_3"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:7_1_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:7_1_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E7:7_1_1_3"/>
 ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> holds <br/><font color="Olive">q</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> ) )
 
<b>by </b><i><a class="ref" href="lattice3.html#D10">LATTICE3:def 10</a></i>;<br/>
<a NAME="E8:7_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D13">LATTICE3:def 13</a></i>;<br/>
<a NAME="E9:7_1_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E10:7_1_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K10">"\/"</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 ;<br/>


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

<b>thus </b><a NAME="E4:7_1_1"/>
 <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font> is <a href="waybel_4.html#V4" target="_self">auxiliary(iv)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_4.html#D7" target="_self">WAYBEL_4:def 7</a><br/>

<a NAME="E1:7_1_1_4"/>
 <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<a NAME="E2:7_1_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E3:7_1_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel_4.html#K2" target="_self">IntRel</a> <font color="Maroon">L</font>
 ;<br/>


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


</div>
