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<b>let </b><font color="Maroon" title="c1">T1</font>, <font color="Maroon" title="c2">T2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E48">A1</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T1</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T2</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font> #)
 ;<br/>

<b>let </b><font color="Maroon" title="c3">N1</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T1</font>;<br/><b>let </b><font color="Maroon" title="c4">N2</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">T2</font>;<br/>



<b>assume </b><a NAME="E2:79"/><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c3">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">p1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T1</font>;<br/><b>let </b><font color="Maroon" title="c6">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T2</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:79"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font>
 <b>and </b><br/><a NAME="E4:79"/><i><font color="Green" title="E51">A4</font></i>: 
<font color="Maroon" title="c5">p1</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">N1</font>
 ;<br/>

<a NAME="E5:79"/>
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">p2</font> holds  <font color="Maroon" title="c4">N2</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="79_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">p2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c8">W</font> = <font color="Maroon" title="c7">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:79"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E2:79_1"/>
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">G</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T2</font><b> such that </b><br/><a NAME="E4:79_1"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c9">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:79_1"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c9">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">V</font>
 <b>and </b><br/><a NAME="E6:79_1"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c6">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">G</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">H</font> = <font color="Maroon" title="c9">G</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T1</font> <b>by </b><i><a class="txt" href="waybel11.html#E1:79"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E8:79_1"/>
<font color="Maroon" title="c9">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">T2</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E4:79_1"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:79_1"/><b>then </b>
<font color="Maroon" title="c10">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:79"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:79_1"/><b>then </b>
<font color="Maroon" title="c5">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c8">W</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E3:79"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="waybel11.html#E5:79_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="waybel11.html#E6:79_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">W</font> = <font color="Maroon" title="c7">V</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c5">p1</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E12:79_1"/>
<font color="Maroon" title="c4">N2</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c7">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:79_1_1"/>
<font color="Maroon" title="c3">N1</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c11">W</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E4:79"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font><b> such that </b><br/><a NAME="E3:79_1_1"/><i><font color="Green" title="E55">A8</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="c3">N1</font>  st <font color="Maroon" title="c12">i</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="c3">N1</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="c11">W</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>reconsider </b><font color="Maroon" title="c13">ii</font> = <font color="Maroon" title="c12">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font> <b>by </b><i><a class="txt" href="waybel11.html#E2:79"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">ii</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="c14">jj</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N2</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c14">jj</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">N1</font> <b>by </b><i><a class="txt" href="waybel11.html#E2:79"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:79_1_1"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c13">ii</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">jj</font>
 ;<br/>

<a NAME="E7:79_1_1"/>
<font color="Maroon" title="c4">N2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N1</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c15">j</font>
 
<b>by </b><i><a class="txt" href="waybel11.html#E2:79"><i><font color="Green" title="E49">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:79_1_1"/>
<font color="Maroon" title="c4">N2</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">jj</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="waybel11.html#E2:79"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="waybel11.html#E3:79_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="waybel11.html#E6:79_1_1"><i><font color="Green" title="E56">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:79"/>
<font color="Maroon" title="c6">p2</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="c4">N2</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></i>;<br/>


</div>
