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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:37"/>
<font color="Maroon" title="c2">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<a NAME="E2:37"/><b>then </b>
<font color="Maroon" title="c2">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">V</font> where <font color="Olive" title="b1">V</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> : for <font color="Olive" title="b2">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/>for <font color="Olive" title="b3">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b3">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" title="YELLOW_6:def.27">YELLOW_6:def 27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:37"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c2">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V</font>
 <b>and </b><br/><a NAME="E5:37"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/>for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">L</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">N</font>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font> holds <br/><font color="Olive" title="b2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">V</font>
 ;<br/>
<b>let </b><font color="Maroon" title="c4">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D3" title="WAYBEL11:def.3">WAYBEL11:def 3</a><br/>

<b>assume </b><a NAME="E6:37"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U1</font>
 ;<br/>

<a NAME="E7:37"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c4">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K3" title="WAYBEL28:func.3">lim_inf-Convergence</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T27" target="_self" title="WAYBEL28:th.27">Th27</a></i>;<br/>
<a NAME="E8:37"/><b>then </b>
 <a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E4:37"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="waybel28.html#E5:37"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="waybel28.html#E6:37"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span><b> such that </b><br/><a NAME="E10:37"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span>  st <font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U1</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/>
<a NAME="E11:37"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span>
 
;<br/>
<a NAME="E12:37"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T32" title="WAYBEL21:th.32">WAYBEL21:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c6">y1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">y1</font>
;<br/>

<b>thus </b><a NAME="E15:37"/>
<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E11:37"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="waybel21.html#T32" title="WAYBEL21:th.32">WAYBEL21:32</a></i>;<br/>

<b>let </b><font color="Maroon" title="c8">x1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E16:37"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D</font>
 <b>and </b><br/><a NAME="E17:37"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">y1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span> <b>by </b><i><a class="txt" href="waybel28.html#E16:37"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="waybel21.html#T32" title="WAYBEL21:th.32">WAYBEL21:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span> ;<br/>
<a NAME="E20:37"/>
 <a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font> is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="waybel21.html#T32" title="WAYBEL21:th.32">WAYBEL21:32</a></i>;<br/>
<a NAME="E21:37"/><b>then </b><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E17:37"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/>
<span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">x</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K4" title="WAYBEL17:func.4">Net-Str</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="waybel_0.html#D8" title="WAYBEL_0:def.8">WAYBEL_0:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="waybel21.html#T32" title="WAYBEL21:th.32">WAYBEL21:32</a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="waybel28.html#E16:37"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E23:37"/>
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U1</font>
 <b>by </b><i><a class="txt" href="waybel28.html#E10:37"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="waybel28.html#E21:37"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>


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