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<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:36"/>
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of  <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font> holds   <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Olive">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</font> be    <a href="yellow_6.html#M2">subnet</a> of  <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font>;<br/>

<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>
<b>set </b><font color="Maroon">f</font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E25</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:36_1"/>
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<a NAME="E5:36_1"/><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:36_1"/>
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>
 
;<br/>
<a NAME="E7:36_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T32">WAYBEL21:32</a></i>;<br/>
<b>then </b><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="yellow_6.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel21.html#T32">WAYBEL21:32</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="txt" href="waybel28.html#E3:36_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/>
<a NAME="E9:36_1"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#T24">YELLOW_2:24</a></i>;<br/>
<a NAME="E10:36_1"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E11:36_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E3:36_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:36_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E5:36_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_4.html#T2">YELLOW_4:2</a></i>;<br/>
<a NAME="E13:36_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E14:36_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E15:36_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D2">WAYBEL_9:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36"/><b>then </b>
(  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">p</font> being <a href="waybel28.html#V1" target="_self">greater_or_equal_to_id</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> holds   <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font></span>)</span> <a href="waybel28.html#K2" target="_self">*</a> <font color="Olive">p</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="waybel17.html#T10">WAYBEL17:10</a></i>;<br/>
<b>hence </b><a NAME="E3:36"/>
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of  <a href="waybel17.html#K4">Net-Str</a> <font color="Maroon">D</font> holds   <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div>
