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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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">T</font>;<br/><b>let </b><font color="Maroon" title="c3">Y</font> be    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>;<br/>





<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">Y</font>,<font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E2:93"/><i><font color="Green" title="E51">A1</font></i>: 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c3">Y</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="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E3:93"/><i><font color="Green" title="E52">A2</font></i>: 
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="c2">N</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="c3">Y</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="c3">Y</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="c4">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" target="_self" title="YELLOW_6:def.12">Def12</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">x</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="E4:93"/><i><font color="Green" title="E53">A3</font></i>: 
<font color="Maroon" title="c5">x</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="c2">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E6:93"/>
for <font color="Olive" title="b1">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">p</font> holds  <font color="Maroon" title="c3">Y</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">V</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" title="c7">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">p</font>;<br/>

<a NAME="E1:93_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E4:93"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">ii</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="E3:93_1"/><i><font color="Green" title="E54">A4</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">ii</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> <font color="Maroon" title="c7">V</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/>
<b>consider </b><font color="Maroon" title="c9">n</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Y</font><b> such that </b><br/><a NAME="E5:93_1"/><i><font color="Green" title="E55">A5</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="c3">Y</font>  st <font color="Maroon" title="c9">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c8">ii</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:93"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">n</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="c10">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Y</font>;<br/>

<b>assume </b><a NAME="E6:93_1"/>
<font color="Maroon" title="c9">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 ;<br/>

<a NAME="E7:93_1"/><b>then </b><i><font color="Green" title="E56">A6</font></i>: 
<font color="Maroon" title="c8">ii</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c10">j</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E5:93_1"><i><font color="Green" title="E55">A5</font></i></a></i>;<br/>
<a NAME="E8:93_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a>, <a class="txt" href="yellow_6.html#E2:93"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:93_1"/>
<font color="Maroon" title="c3">Y</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:93_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="yellow_6.html#E7:93_1"><i><font color="Green" title="E56">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:93"/>
<font color="Maroon" title="c5">x</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="c3">Y</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>


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