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

<b>thus </b><a NAME="E1:12_1_1"/>
( <font color="Maroon">M</font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a> implies for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">M</font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a>
 ;<br/>

<a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">M</font> is <a href="tsp_1.html#V1">T_0</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E3:12_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E4:12_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:12_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">MaxADSet</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tex_4.html#D11">TEX_4:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:12_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E8:12_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">C</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">MaxADSet</a> <font color="Olive">a</font></span>)</span> where <font color="Olive">a</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</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">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E10:12_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E11:12_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>
<a NAME="E12:12_1_1_1"/>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="tsp_2.html#D1" target="_self">Def1</a>, <a class="ref" href="tex_4.html#T23">TEX_4:23</a></i>;<br/>
<a NAME="E13:12_1_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E14:12_1_1_1"/>
 ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:12_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<a NAME="E3:12_1_1"/>
for <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b</font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:12_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:12_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:12_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E5:12_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">MaxADSet</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/>
<a NAME="E6:12_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">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="E7:12_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:12_1_1_2"/>
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:12_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">M</font> is <a href="tsp_1.html#V1">T_0</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:12_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:12_1_1_3_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E5:12_1_1_3_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:12_1_1_3_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:12_1_1_3_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E8:12_1_1_3_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font></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_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:12_1_1_3_1"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T34">TEX_4:34</a></i>;<br/><a NAME="E10:12_1_1_3_1"/><b>then </b>
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E11:12_1_1_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font></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="tsp_2.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E12:12_1_1_3_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E13:12_1_1_3_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T32">TEX_4:32</a></i>;<br/><a NAME="E14:12_1_1_3_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/><a NAME="E15:12_1_1_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E16:12_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:12_1_1_3"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12_1_1"/><b>then </b>
 <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:12_1_1"/>
<font color="Maroon">M</font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a>
 <b>by </b><i>, </i>;<br/>


</div>
