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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:46_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">T</font> is <a href="compts_1.html#V2">compact</a>
 ;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">S</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:46_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#T32">WAYBEL_9:32</a></i>;<br/><b>take </b><font color="Maroon">S</font> = <font color="Maroon">S</font>;<br/><b>thus </b><a NAME="E6:46_1"/>
<font color="Maroon">S</font> is <a href="yellow_6.html#V3">convergent</a>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D19">YELLOW_6:def 19</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:46"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>  ex <font color="Olive">S</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Olive">N</font> st <font color="Olive">S</font> is <a href="yellow_6.html#V3">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">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>consider </b><font color="Maroon">S</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E2:46_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S</font> is <a href="yellow_6.html#V3">convergent</a>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">S</font>;<br/><a NAME="E4:46_2"/><i><font color="Green">E55</font></i>: 
 <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D19">YELLOW_6:def 19</a></i>;<br/><a NAME="E5:46_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><a NAME="E7:46_2"/>
<font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#T29">WAYBEL_9:29</a></i>;<br/><b>hence </b><a NAME="E8:46_2"/>
<font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T31">WAYBEL_9:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:46"/>
<font color="Maroon">T</font> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i/>;<br/>


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