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<div class="add">

<b>let </b><font color="Maroon">R</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">Z</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> </span>}</span> ,<font color="Maroon">R</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : verum </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font>;<br/>
<a NAME="E3:65"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:65"/>
not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E5:65"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">jx</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E7:65"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jx</font> </span>}</span> ,<font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E8:65"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">jy</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E10:65"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jy</font> </span>}</span> ,<font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">jx</font> = <font color="Maroon">jx</font>, <font color="Maroon">jy</font> = <font color="Maroon">jy</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> ;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E13:65"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jx</font>
 <b>and </b><br/><a NAME="E14:65"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jy</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">E1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jx</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Ey</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jy</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">E</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ;<br/>
<b>take </b><font color="Maroon">z</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">R</font>;<br/>

<b>thus </b><a NAME="E15:65"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>

<a NAME="E16:65"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jx</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:65_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E3:65_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:65_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> ;<br/>
<a NAME="E6:65_1"/>
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jx</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:65_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jx</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel11.html#E1:65"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:65"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>

<a NAME="E18:65"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jy</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:65_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font><b> such that </b><br/><a NAME="E3:65_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:65_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> ;<br/>
<a NAME="E6:65_2"/>
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jy</font>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:65_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Z</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> : <font color="Olive">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">jy</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel11.html#E1:65"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:65"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>


</div>
