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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="yellow_6.html#V8" target="_self">topological</a>  <a href="yellow_6.html#M4" target="_self">Convergence-Class</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:119"/><i><font color="Green">E72</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:119_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E2:119_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E3:119_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/><b>assume </b><a NAME="E4:119_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E5:119_1"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T10">WAYBEL_0:10</a></i>;<br/><a NAME="E6:119_1"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119"><i><font color="Green">E72</font></i></a></i>;<br/><a NAME="E7:119_1"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E8:119_1"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:119_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="yellow_6.html#E3:119_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="yellow_6.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E9:119_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119"><i><font color="Green">E72</font></i></a>, <a class="ref" href="yellow_6.html#T10" target="_self">Th10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:119"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:119_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E2:119_2"/><b>then </b><i><font color="Green">E74</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T10" target="_self">Th10</a></i>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E3:119_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E4:119_2"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:119"><i><font color="Green">E73</font></i></a>, <a class="txt" href="yellow_6.html#E2:119_2"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E5:119_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:119"><i><font color="Green">E72</font></i></a>, <a class="ref" href="waybel_0.html#T10">WAYBEL_0:10</a></i>;<br/><a NAME="E6:119_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>hence </b><a NAME="E7:119_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E5:119"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T50" target="_self">Th50</a></i>;<br/>
<b>hence </b><a NAME="E6:119"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


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