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

<b>let </b><font color="Maroon">T</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="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="waybel21.html#M2" target="_self">Embedding</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font>;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Maroon">e</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">j'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">j'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">j</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j'</font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</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">a<sub>1</sub></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">a<sub>1</sub></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">T</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">a<sub>1</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E2:48"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">N</font>),<font color="Maroon">T</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">M</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</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="E1:48_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">M</font>)
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">t</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">M</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">M</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">T</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> ;<br/>
<b>set </b><font color="Maroon">j'</font> = <font color="Maroon">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">X</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">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</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">M</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="E5:48_1"/>
<font color="Maroon">t</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">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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">T</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="E1:48_1_1"/>
<font color="Maroon">x</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">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:48_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E6:48_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="txt" href="waybel21.html#E3:48_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E7:48_1_1"/>
<font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">M</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">M</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/>
<a NAME="E8:48_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="waybel21.html#E3:48_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1_1"/>
<font color="Maroon">t</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel21.html#E3:48_1_1"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:48_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</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">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E7:48_1"/>
 <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">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">T</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">N</font>)
 
;<br/>
<a NAME="E8:48_1"/><b>then </b>
 <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">e</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> </span>}</span> ,<font color="Maroon">T</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">N</font>),<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1"/>
<font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">N</font>),<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:48"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">N</font>),<font color="Maroon">T</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">M</font>),<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E4:48"/><b>then </b>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">M</font>),<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel11.html#D6">WAYBEL11:def 6</a></i>;<br/>
<a NAME="E5:48"/><b>then </b>
(  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font> &amp;  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="waybel21.html#D1" target="_self">Def1</a>, <a class="ref" href="waybel11.html#D6">WAYBEL11:def 6</a></i>;<br/>
<b>hence </b><a NAME="E6:48"/>
 <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div>
