<?xml version="1.0"?>
<div class="add">

<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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:81"/>
(  not  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> or 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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> ) or 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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>assume </b><a NAME="E2:81_1"/><i><font color="Green">E54</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;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/><b>thus </b><a NAME="E1:81_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:81_1_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_4.html#T50" target="_self">Th50</a></i>;<br/><b>hence </b><a NAME="E3:81_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:81_1_1_1"/>
not  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:81_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:81_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E5:81_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E4:81_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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="tex_4.html#E4:81_1_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E7:81_1_1_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E5:81_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="tex_4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E8:81_1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_4.html#T51" target="_self">Th51</a></i>;<br/><a NAME="E9:81_1_1_1"/>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E4:81_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:81_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E11:81_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E12:81_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E8:81_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/><a NAME="E13:81_1_1_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_4.html#T50" target="_self">Th50</a></i>;<br/><a NAME="E14:81_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><a NAME="E15:81_1_1_1"/><b>then </b>
not  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E2:81_1"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E16:81_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:81_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E19:81_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E18:81_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</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="tex_4.html#E18:81_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E21:81_1_1_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tex_4.html#E19:81_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tex_4.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E22:81_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_4.html#T51" target="_self">Th51</a></i>;<br/><a NAME="E23:81_1_1_1"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E18:81_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:81_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E25:81_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E26:81_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E22:81_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/><a NAME="E27:81_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_4.html#E12:81_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E28:81_1_1_1"/><b>then </b>
(  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T51" target="_self">Th51</a></i>;<br/><b>hence </b><a NAME="E29:81_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_4.html#E1:81_1"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:81_1_1"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:81_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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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

<b>assume </b><a NAME="E2:81"/><i><font color="Green">E53</font></i>: 
( 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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> ) or 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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>

<b>assume </b><a NAME="E3:81"/>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:81"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T24" target="_self">Th24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_2_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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> ) or 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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="tex_4.html#E2:81"><i><font color="Green">E53</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_1"><b>suppose </b></a><a NAME="E1:81_2_1_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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><b>then 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="E3:81_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:81_2_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:81_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E6:81_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E5:81_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E7:81_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_4.html#E4:81"><i><font color="Green">E54</font></i></a>, <a class="txt" href="tex_4.html#E4:81_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_2_1_2"><b>suppose </b></a><a NAME="E1:81_2_1_2"/>
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="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><div class="add"><b>then 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="E3:81_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:81_2_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:81_2_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E6:81_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tex_4.html#E4:81_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E7:81_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_4.html#E4:81"><i><font color="Green">E54</font></i></a>, <a class="txt" href="tex_4.html#E5:81_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:81"/>
contradiction
 ;<br/>


</div>
