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

<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="compts_1.html#NV6" title="COMPTS_1:NV.6">Hausdorff</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c4">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:89"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 <b>and </b><br/><a NAME="E2:89"/><i><font color="Green" title="E43">A2</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 <font color="Maroon" title="c1">S</font> holds  <font color="Olive" title="b1">x</font> <a href="waybel_1.html#K4" title="WAYBEL_1:func.4">"/\"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E3:89"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">N</font> is <a href="waybel_0.html#V10" title="WAYBEL_0:attr.10">eventually-directed</a>
 <b>and </b><br/><a NAME="E4:89"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E5:89"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">iJ</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:89"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c6">iJ</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font>
 <b>and </b><br/><a NAME="E8:89"/><i><font color="Green" title="E47">A6</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">iJ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">i1</font> = <font color="Maroon" title="c6">iJ</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="txt" href="waybel_9.html#E7:89"><i><font color="Green" title="E46">A5</font></i></a></i>;<br/>
<a NAME="E10:89"/><i><font color="Green" title="E48">A7</font></i>: 
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : <font color="Olive" title="b1">z</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green" title="E5">Lm1</font></i></a></i>;<br/>
<a NAME="E11:89"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c8">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E2:89_1"/><i><font color="Green" title="E50">A9</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c8">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E3:89"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="waybel_0.html#T11" title="WAYBEL_0:th.11">WAYBEL_0:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">j</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon" title="c9">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>assume </b><a NAME="E3:89_1"/>
<font color="Maroon" title="c8">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 ;<br/>

<a NAME="E4:89_1"/><b>then </b>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:89_1"><i><font color="Green" title="E50">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:89_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18" title="WAYBEL_0:th.18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:89"/>
 <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:89"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="waybel_9.html#T27" target="_self" title="WAYBEL_9:th.27">Th27</a></i>;<br/>
<a NAME="E13:89"/><b>then </b><i><font color="Green" title="E50">A10</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E15:89"/><i><font color="Green" title="E51">A11</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c8">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_9.html#E11:89"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = <font color="Maroon" title="c2">N</font> <a href="waybel_9.html#K6" title="WAYBEL_9:func.6">|</a> <font color="Maroon" title="c8">t</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<a NAME="E17:89"/><i><font color="Green" title="E52">A12</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:89_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:89_2"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">A</font>
 <b>and </b><br/><a NAME="E4:89_2"/><i><font color="Green" title="E54">A14</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font> <b>by </b><i><a class="txt" href="waybel_9.html#E3:89_2"><i><font color="Green" title="E53">A13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E7:89_2"/><i><font color="Green" title="E55">A15</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font>
 <b>and </b><br/><a NAME="E8:89_2"/><i><font color="Green" title="E56">A16</font></i>: 
<font color="Maroon" title="c8">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">y</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D7" target="_self" title="WAYBEL_9:def.7">Def7</a></i>;<br/>
<a NAME="E9:89_2"/><i><font color="Green" title="E57">A17</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E15:89"><i><font color="Green" title="E51">A11</font></i></a>, <a class="txt" href="waybel_9.html#E8:89_2"><i><font color="Green" title="E56">A16</font></i></a></i>;<br/>
<a NAME="E10:89_2"/>
<font color="Maroon" title="c9">A</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">y</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E7:89_2"><i><font color="Green" title="E55">A15</font></i></a>, <a class="ref" href="waybel_9.html#T16" target="_self" title="WAYBEL_9:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E11:89_2"/>
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_9.html#E4:89_2"><i><font color="Green" title="E54">A14</font></i></a>, <a class="txt" href="waybel_9.html#E9:89_2"><i><font color="Green" title="E57">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:89"/>
<font color="Maroon" title="c3">c</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c9">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">O</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">c</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D9" target="_self" title="WAYBEL_9:def.9">WAYBEL_9:def 9</a><br/><b>let </b><font color="Maroon" title="c11">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a><br/>



<b>consider </b><font color="Maroon" title="c12">i2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E2:89_3"/><i><font color="Green" title="E53">A18</font></i>: 
<font color="Maroon" title="c12">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font>
 <b>and </b><br/><a NAME="E3:89_3"/><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c8">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#D7" target="_self" title="WAYBEL_9:def.7">Def7</a></i>;<br/>
<a NAME="E4:89_3"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c10">O</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:89"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="waybel_9.html#D9" target="_self" title="WAYBEL_9:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">j2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E6:89_3"/><i><font color="Green" title="E55">A20</font></i>: 
<font color="Maroon" title="c12">i2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font>
 <b>and </b><br/><a NAME="E7:89_3"/><i><font color="Green" title="E56">A21</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c13">j2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">O</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E8:89_3"/>
<font color="Maroon" title="c8">t</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:89_3"><i><font color="Green" title="E54">A19</font></i></a>, <a class="txt" href="waybel_9.html#E6:89_3"><i><font color="Green" title="E55">A20</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">j</font> = <font color="Maroon" title="c13">j2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font> <b>by </b><i><a class="ref" href="waybel_9.html#D7" target="_self" title="WAYBEL_9:def.7">Def7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">j</font>
;<br/>

<a NAME="E10:89_3"/>
<font color="Maroon" title="c9">A</font> is  <a href="yellow_6.html#V2" title="YELLOW_6:attr.2">full</a>  <a href="yellow_6.html#M1" title="YELLOW_6:mode.1">SubNetStr</a> of <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T14" target="_self" title="WAYBEL_9:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E11:89_3"/>
<font color="Maroon" title="c11">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E2:89_3"><i><font color="Green" title="E53">A18</font></i></a>, <a class="txt" href="waybel_9.html#E6:89_3"><i><font color="Green" title="E55">A20</font></i></a>, <a class="ref" href="yellow_6.html#T21" title="YELLOW_6:th.21">YELLOW_6:21</a></i>;<br/>

<b>thus </b><a NAME="E12:89_3"/>
<font color="Maroon" title="c9">A</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">O</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E7:89_3"><i><font color="Green" title="E56">A21</font></i></a>, <a class="ref" href="waybel_9.html#T16" target="_self" title="WAYBEL_9:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c10">M</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c9">A</font><b> such that </b><br/><a NAME="E20:89"/><i><font color="Green" title="E53">A22</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c10">M</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T32" target="_self" title="WAYBEL_9:th.32">Th32</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c10">M</font>,<font color="Maroon" title="c9">A</font><b> such that </b><br/><a NAME="E22:89"/><i><font color="Green" title="E54">A23</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c10">M</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">A</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">f</font>
 <b>and </b><br/><a NAME="E23:89"/>
for <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">A</font>  ex <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">M</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">M</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b3">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">R</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c10">M</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E25:89"/>
<font color="Maroon" title="c12">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c9">A</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E22:89"><i><font color="Green" title="E54">A23</font></i></a>, <a class="ref" href="relat_1.html#T45" title="RELAT_1:th.45">RELAT_1:45</a></i>;<br/>
<a NAME="E26:89"/><b>then </b>
<font color="Maroon" title="c12">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E17:89"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E27:89"/><b>then </b><i><font color="Green" title="E55">A24</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c12">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E28:89"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c12">R</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E20:89"><i><font color="Green" title="E53">A22</font></i></a>, <a class="ref" href="waybel_9.html#T24" target="_self" title="WAYBEL_9:th.24">Th24</a></i>;<br/>
<a NAME="E29:89"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E13:89"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="waybel_9.html#E27:89"><i><font color="Green" title="E55">A24</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E31:89"/><i><font color="Green" title="E56">A25</font></i>: 
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z</font> &amp; <font color="Maroon" title="c13">z</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font> )
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:89"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/>
<a NAME="E32:89"/>
<font color="Maroon" title="c4">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">i1</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:89"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="waybel_9.html#E31:89"><i><font color="Green" title="E56">A25</font></i></a>, <a class="ref" href="yellow_0.html#T25" title="YELLOW_0:th.25">YELLOW_0:25</a></i>;<br/>
<b>hence </b><a NAME="E33:89"/>
<font color="Maroon" title="c4">d</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E8:89"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/>


</div>
