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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="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</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="waybel_7.html#V3" target="_self">ultra</a> <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:43"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="waybel_7.html#R1" target="_self">is_a_cluster_point_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="waybel_7.html#R2" target="_self">is_a_convergence_point_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D4" target="_self">WAYBEL_7:def 4</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5" target="_self">WAYBEL_7:def 5</a><br/>

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:43_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_7.html#V2" target="_self">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E4:43_1"/><b>then </b>
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>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="waybel_7.html#E2:43_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_7.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:43_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</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>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#R1">misses</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>5</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E1:43_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel_7.html#E2:43_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>hence </b><a NAME="E6:43_1"/>
contradiction
 ;<br/>


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

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5" target="_self">WAYBEL_7:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D4" target="_self">WAYBEL_7:def 4</a><br/>

<b>assume </b><a NAME="E3:43"/>
( <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<a NAME="E4:43"/><b>then </b><i><font color="Green">E29</font></i>: 
<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="waybel_7.html#E2:43"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:43"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel_7.html#E4:43"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E7:43"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a>, <a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/>
<a NAME="E8:43"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K8">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_7.html#E4:43"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel_7.html#E5:43"><i><font color="Green">E30</font></i></a>, <a class="ref" href="waybel_7.html#T8" target="_self">Th8</a>, <a class="ref" href="waybel_0.html#T41">WAYBEL_0:41</a></i>;<br/>
<b>hence </b><a NAME="E9:43"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div>
