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

<b>let </b><font color="Maroon">X</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">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:91"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a> or <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<a NAME="E2:91"/><i><font color="Green">E20</font></i>: 
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:91"/>
<span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/>
not <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:91_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font></span>)</span>
 <b>and </b><br/><a NAME="E4:91_1"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1"/>
( <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a> or <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:91_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:91_1_1_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, </i>;<br/><a NAME="E5:91_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:91_1_1_1_1"/><b>then </b>
<font color="Maroon">x</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#K4" target="_self">MaxADSet</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:91_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E9:91_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">D</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#K4" target="_self">MaxADSet</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</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">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:91_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E12:91_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E13:91_1_1_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, <a class="txt" href="tex_4.html#E4:91_1_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E14:91_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</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">b</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E3:91_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E15:91_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E16:91_1_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#D7" target="_self">Def7</a>, <a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:91_1_1_1_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:91_1_1_1_1"/><b>then </b>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#T22" target="_self">Th22</a>, <a class="ref" href="tex_4.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E19:91_1_1_1_1"/><b>then </b>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E20:91_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self">Th14</a>, <a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E3:91_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:91_1_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T15" target="_self">Th15</a>, </i>;<br/><a NAME="E5:91_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:91_1_1_1_2"/><b>then </b>
<font color="Maroon">x</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#K4" target="_self">MaxADSet</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">D</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:91_1_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E9:91_1_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">D</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#K4" target="_self">MaxADSet</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</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">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E11:91_1_1_1_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E12:91_1_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E13:91_1_1_1_2"/><i><font color="Green">E118</font></i>: 
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a>, </i>;<br/><a NAME="E14:91_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</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">b</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E3:91_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E15:91_1_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E16:91_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tex_4.html#D7" target="_self">Def7</a>, <a class="ref" href="tex_4.html#T20" target="_self">Th20</a>, <a class="ref" href="tex_4.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:91_1_1_1_2"/>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:91_1_1_1_2"/><b>then </b>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#T22" target="_self">Th22</a>, <a class="ref" href="tex_4.html#T23" target="_self">Th23</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E19:91_1_1_1_2"/><b>then </b>
 <a href="tex_4.html#K4" target="_self">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E20:91_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tex_4.html#T14" target="_self">Th14</a>, <a class="ref" href="tex_4.html#T17" target="_self">Th17</a>, <a class="ref" href="tex_4.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:91_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:91"/>
 <a href="tex_4.html#K3" target="_self">MaxADSet</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" target="_self">MaxADSet</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_4.html#T12" target="_self">Th12</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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