<?xml version="1.0"?>
<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">N</font> be  <a href="yellow_6.html#V1">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font>;<br/>



<b>set </b><font color="Maroon">X</font> = <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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">N</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">N</font>;<br/>
<a NAME="E2:55"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">N</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">N</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:55_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">b</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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j0</font> </span>}</span> ,<font color="Maroon">R</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">j0</font> = <font color="Maroon">j0</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>consider </b><font color="Maroon">i0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:55_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j0</font>
 <b>and </b><br/><a NAME="E7:55_1"/>
<font color="Maroon">i0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j0</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E8:55_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j0</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> = 
 <a href="yellow_6.html#K5">the_value_of</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
<br/>.= 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
;<br/>
<b>hence </b><a NAME="E10:55_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:55"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">N</font> : verum </span>}</span>  holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ;<br/>
<a NAME="E1:55_2"/><i><font color="Green">E88</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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> <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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">N</font> : verum </span>}</span> 
 
;<br/>
<b>assume </b><a NAME="E2:55_2"/>
<font color="Maroon">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</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">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">N</font> : verum </span>}</span> 
 ;<br/>

<a NAME="E3:55_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E4:55_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:55_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</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">i0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:55_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font>
 <b>and </b><br/><a NAME="E4:55_2_1"/>
<font color="Maroon">i0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
 <a href="yellow_6.html#K5">the_value_of</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
<br/>.= 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
;<br/>
<b>hence </b><a NAME="E6:55_2_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:55_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">b</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span>  holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>

<b>consider </b><font color="Maroon">i0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E2:55_2_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">i0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E3:55_2_2"/>
<font color="Maroon">i0</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E4:55_2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:55_2_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> = 
 <a href="yellow_6.html#K5">the_value_of</a> <font color="Maroon">N</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
<br/>.= 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i0</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
;<br/>
<b>hence </b><a NAME="E7:55_2_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="waybel11.html#E1"><i><font color="Green">Lemma34</font></i></a>, , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:55_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b> <a href="yellow_6.html#K5">the_value_of</a> <font color="Maroon">N</font> = 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25">YELLOW_6:25</a></i>
<br/>.= 
 <a href="waybel11.html#K1" target="_self">lim_inf</a> <font color="Maroon">N</font>
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>
;<br/>


</div>
