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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#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <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><a NAME="E1:63"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>assume </b><a NAME="E2:63"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:63"/>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:63_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:63_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:63_1"><i><font color="Green">E44</font></i></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="E5:63_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:63_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:63_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E9:63_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>7</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="tex_4.html#E6:63_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E10:63_1"/><i><font color="Green">E49</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E5:63_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="tex_4.html#E8:63_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tex_4.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E11:63_1"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="tex_4.html#E1:63"><i><font color="Green">E42</font></i></a>, <a class="txt" href="tex_4.html#E2:63"><i><font color="Green">E43</font></i></a>, <a class="txt" href="tex_4.html#E9:63_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E12:63_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E10:63_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:63_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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