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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="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="yellow_6.html#V8" title="YELLOW_6:attr.8">topological</a>  <a href="yellow_6.html#M4" title="YELLOW_6:mode.4">Convergence-Class</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>;<br/>





<b>set </b><font color="Maroon" title="c4">CC</font> =  <a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font>;<br/>
<a NAME="E1:119"/><i><font color="Green" title="E69">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self" title="YELLOW_6:def.27">Def27</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:119_1"/><i><font color="Green" title="E70">A2</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c6">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>assume </b><a NAME="E2:119_1"/><i><font color="Green" title="E71">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">N</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/><a NAME="E3:119_1"/><i><font color="Green" title="E72">A4</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119_1"><i><font color="Green" title="E70">A2</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/><b>assume </b><a NAME="E4:119_1"/>
<font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">S</font>
 ;<br/><a NAME="E5:119_1"/><b>then </b>
not <font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="yellow_6.html#K14" title="YELLOW_6:func.14">ConvergenceSpace</a> <font color="Maroon" title="c2">C</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="waybel_0.html#T10" title="WAYBEL_0:th.10">WAYBEL_0:10</a></i>;<br/><a NAME="E6:119_1"/><b>then </b>
not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:119_1"><i><font color="Green" title="E71">A3</font></i></a>, <a class="txt" href="yellow_6.html#E3:119_1"><i><font color="Green" title="E72">A4</font></i></a>, <a class="ref" href="yellow_6.html#T50" target="_self" title="YELLOW_6:th.50">Th50</a></i>;<br/><b>hence </b><a NAME="E7:119_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:119"/><i><font color="Green" title="E70">A5</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">N</font>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font> &amp; <font color="Olive" title="b2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">S</font> holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:119_2"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><a NAME="E2:119_2"/><b>then </b><i><font color="Green" title="E71">A6</font></i>: 
not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>let </b><font color="Maroon" title="c6">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>assume </b><a NAME="E3:119_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">N</font>,<font color="Maroon" title="c5">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 ;<br/><a NAME="E4:119_2"/><b>then </b>
not <font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:119"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="yellow_6.html#E2:119_2"><i><font color="Green" title="E71">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E5:119_2"/>
<font color="Maroon" title="c6">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119"><i><font color="Green" title="E69">A1</font></i></a>, <a class="ref" href="waybel_0.html#T10" title="WAYBEL_0:th.10">WAYBEL_0:10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:119"/><b>then </b>
<font color="Maroon" title="c3">S</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T50" target="_self" title="YELLOW_6:th.50">Th50</a></i>;<br/>
<b>hence </b><a NAME="E6:119"/>
<font color="Maroon" title="c3">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


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