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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:93"/><i><font color="Green">E54</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:93"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:93"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E6:93"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font> holds <font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E1:93_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E4:93"><i><font color="Green">E56</font></i></a>, <a class="ref" href="yellow_6.html#D18" target="_self">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:93_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:93_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:93"><i><font color="Green">E55</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E6:93_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E7:93_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E5:93_1"><i><font color="Green">E58</font></i></a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="yellow_6.html#E2:93"><i><font color="Green">E54</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:93_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:93_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="yellow_6.html#E7:93_1"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:93"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18" target="_self">Def18</a></i>;<br/>


</div>
