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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>assume </b><a NAME="E1:45"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR3" target="_self">is_T3</a> 
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V6">locally-compact</a>
 ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:45_1"/>
ex <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 <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6">compact</a> )
 <b>by </b><i><a class="txt" href="yellow_8.html#E1:45_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel_3.html#D9">WAYBEL_3:def 9</a></i>;<br/><b>hence </b><a NAME="E3:45_1"/>
ex <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 <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="compts_1.html#V6">compact</a> )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:45"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9">WAYBEL_3:def 9</a><br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E4:45"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:45"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:45"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:45"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="yellow_8.html#E3:45"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E9:45"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E4:45"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow_8.html#E5:45"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<a NAME="E10:45"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E7:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:45"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T46">TOPS_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:45"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E14:45"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E15:45"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E1:45"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_8.html#T36" target="_self">Th36</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E16:45"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E15:45"><i><font color="Green">E36</font></i></a></i>;<br/>

<a NAME="E17:45"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E18:45"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E14:45"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E19:45"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E20:45"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_8.html#E14:45"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E21:45"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="yellow_8.html#E8:45"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow_8.html#E13:45"><i><font color="Green">E34</font></i></a>, <a class="ref" href="compts_1.html#T18">COMPTS_1:18</a></i>;<br/>


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