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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
(  not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:40_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></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="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:40_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:40_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <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/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:40_1_1_1_1_2_1"/>
<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="E2:40_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="topgen_1.html#E1:40_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:40_1_1_1_1_2_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="topgen_1.html#E4:40_1_1_1_1_2_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E6:40_1_1_1_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:40_1_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topgen_1.html#E4:40_1_1_1_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:40_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topgen_1.html#T19" target="_self">Th19</a>, <a class="txt" href="topgen_1.html#E1:40_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E4:40_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon">c<sub>3</sub></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="E3:40_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E4:40_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="topgen_1.html#T30" target="_self">Th30</a>, <a class="txt" href="topgen_1.html#E1:40_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="topgen_1.html#E3:40_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2_1"><b>suppose </b></a><a NAME="E1:40_1_1_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:40_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><b>hence </b><a NAME="E3:40_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E1:40_1_1_2_1"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_2_2"><b>suppose </b></a><a NAME="E1:40_1_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E4:40_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"> <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="topgen_1.html#E1:40_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topgen_1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 

<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topgen_1.html#E1:40_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="topgen_1.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:40_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
