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<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">X0</font> be   <a href="tsp_2.html#NM1">maximal_Kolmogorov_subspace</a> of <font color="Maroon">X</font>;<br/>



<b>reconsider </b><font color="Maroon">M</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E2:26"/><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>
 
<b>by </b><i><a class="ref" href="tsp_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>thus </b><a NAME="E3:26"/>
( <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> implies ( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">G</font> &amp;  ex <font color="Olive">G0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> st <br/>( <font color="Olive">G0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tex_4.html#T58">TEX_4:58</a></i>;<br/>

<b>reconsider </b><font color="Maroon">G0</font> = <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">M</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G0</font> = <font color="Maroon">G0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font> ;<br/>
<b>take </b>
<font color="Maroon">G0</font>
;<br/>

<b>thus </b><a NAME="E5:26_1"/>
<font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_2.html#E4:26"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tsp_1.html#D1">TSP_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:26_1"/>
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>


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

<b>assume </b><a NAME="E4:26"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">G</font>
 ;<br/>

<b>given </b><font color="Maroon">G0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X0</font><b> such that </b><a NAME="E5:26"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">G0</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">G0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>

<b>set </b><font color="Maroon">E</font> = <font color="Maroon">G0</font>;<br/>
<a NAME="E7:26"/>
( <font color="Maroon">G0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">G0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:26"/><i><font color="Green">E23</font></i>: 
 <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="txt" href="tsp_2.html#E2:26"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E10:26"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">E</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E4:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="tsp_2.html#E5:26"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E11:26"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </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="E1:26_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:26_2_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> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E4:26_2_1"/><b>then </b>
 <a href="tex_4.html#K3">MaxADSet</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> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E4:26"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tex_4.html#T36">TEX_4:36</a></i>;<br/>
<a NAME="E5:26_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T30">TEX_4:30</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="E7:26_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E8:26_2_1"/><i><font color="Green">E29</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">a</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>, </i>;<br/>
<a NAME="E9:26_2_1"/><i><font color="Green">E30</font></i>: 
<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">a</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:26_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:26_2_1"/><b>then </b>
<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> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E10:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:26_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E12:26_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:26"><i><font color="Green">E21</font></i></a>, <a class="txt" href="tsp_2.html#E1:26_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E13:26_2_1"/><b>then </b>
<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> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E14:26_2_1"/><b>then </b>
 <a href="tex_4.html#K3">MaxADSet</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> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T33">TEX_4:33</a></i>;<br/>
<a NAME="E15:26_2_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T30">TEX_4:30</a></i>;<br/>
<a NAME="E16:26_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E7:26_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E17:26_2_1"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T23">TEX_4:23</a></i>;<br/>
<a NAME="E18:26_2_1"/><b>then </b>
<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">b</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/>
<a NAME="E19:26_2_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">b</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E20:26_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E8:26_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_2"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K3">MaxADSet</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:26"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_2.html#E4:26"><i><font color="Green">E19</font></i></a>, <a class="txt" href="tsp_2.html#E6:26"><i><font color="Green">E22</font></i></a>, <a class="txt" href="tsp_2.html#E9:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tex_4.html#T37">TEX_4:37</a></i>;<br/>


</div>
