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<b>let </b><font color="Maroon">S</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</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:88"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<b>consider </b><font color="Maroon">M</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:88"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:88"/><i><font color="Green">E40</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E8:88"/>
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E10:88"/>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<a NAME="E11:88"/><b>then </b>
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:88"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E13:88"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel_9.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E14:88"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:88"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>


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