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



<b>thus </b><a NAME="E1:6"/>
 <a href="topgen_1.html#K2">Der</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2">Der</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:6_1"/>
<font color="Maroon">x</font> <a href="topgen_1.html#R1">is_an_accumulation_point_of</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="topgen_1.html#D3">TOPGEN_1:def 3</a></i>;<br/>
<a NAME="E3:6_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D2">TOPGEN_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:6_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:6"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 ;<br/>
<a NAME="E5:6"/>
<font color="Maroon">z</font> <a href="topgen_1.html#R1">is_an_accumulation_point_of</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="topgen_1.html#D2">TOPGEN_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:6"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2">Der</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="topgen_1.html#D3">TOPGEN_1:def 3</a></i>;<br/>


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