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





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be  <a href="tex_2.html#V1">proper</a> <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="yellow19.html#K1" target="_self">NeighborhoodSystem</a> <font color="Maroon">x</font>
 ;<br/>

<b>set </b><font color="Maroon">N</font> =  <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font>;<br/>
<a NAME="E3:26"/><i><font color="Green">E37</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Olive">f</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a><br/>

<a NAME="E4:26"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">f</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> <b>by </b><i>, </i>;<br/>
<a NAME="E8:26"/>
<font color="Maroon">f</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="connsp_2.html#T34">CONNSP_2:34</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:26"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E12:26"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">j</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">j</font>
;<br/>

<a NAME="E14:26"/>
( <font color="Maroon">i</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E15:26"/>
( <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/>


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