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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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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>consider </b><font color="Maroon" title="c3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c3">x</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>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:46_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="yellow19.html#T35" target="_self" title="YELLOW19:th.35">Th35</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">S</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E5:46_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c3">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="c4">S</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:46_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="waybel_9.html#T32" title="WAYBEL_9:th.32">WAYBEL_9:32</a></i>;<br/><b>take </b><font color="Maroon" title="c5">S</font> = <font color="Maroon" title="c4">S</font>;<br/><b>thus </b><a NAME="E6:46_1"/>
<font color="Maroon" title="c5">S</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>
 <b>by </b><i><a class="txt" href="yellow19.html#E5:46_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:46"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">S</font> being   <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font> st <font color="Olive" title="b2">S</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>now </b></a><div class="add"><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>consider </b><font color="Maroon" title="c3">S</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E2:46_2"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="yellow_6.html#V3" title="YELLOW_6:attr.3">convergent</a>
 <b>by </b><i><a class="txt" href="yellow19.html#E2:46"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c3">S</font>;<br/><a NAME="E4:46_2"/><i><font color="Green" title="E30">A6</font></i>: 
 <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E2:46_2"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="yellow_6.html#D19" title="YELLOW_6:def.19">YELLOW_6:def 19</a></i>;<br/><a NAME="E5:46_2"/><b>then </b>
<font color="Maroon" title="c4">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">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">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/><b>take </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font>;<br/><a NAME="E7:46_2"/>
<font color="Maroon" title="c6">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E4:46_2"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="waybel_9.html#T29" title="WAYBEL_9:th.29">WAYBEL_9:29</a></i>;<br/><b>hence </b><a NAME="E8:46_2"/>
<font color="Maroon" title="c6">x</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>
 <b>by </b><i><a class="ref" href="waybel_9.html#T31" title="WAYBEL_9:th.31">WAYBEL_9:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46"/>
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="yellow19.html#T35" target="_self" title="YELLOW19:th.35">Th35</a></i>;<br/>


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