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<b>thus </b><a NAME="E1:3_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tsp_1.html#V1">T_0</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</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>2</sub></font> is <a href="tex_4.html#V3">maximal_anti-discrete</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tsp_1.html#V1">T_0</a>
 ;<br/>

<b>let </b><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>assume </b><a NAME="E2:3_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>4</sub></font> =  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>thus </b><a NAME="E3:3_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tex_4.html#V3">maximal_anti-discrete</a>
 <b>by </b><i><a class="ref" href="tex_4.html#T22">TEX_4:22</a></i>;<br/>

<b>thus </b><a NAME="E4:3_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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="tsp_2.html#E1:3_1_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="tsp_2.html#E2:3_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tsp_2.html#D2" target="_self">Def2</a></i>;<br/>


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

<b>assume </b><a NAME="E2:3_1_1"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</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>2</sub></font> is <a href="tex_4.html#V3">maximal_anti-discrete</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>now </b></a><div class="add"><b>let </b><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>assume </b><a NAME="E1:3_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><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="E3:3_1_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="tex_4.html#V3">maximal_anti-discrete</a>
 <b>and </b><br/><a NAME="E4:3_1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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="tsp_2.html#E2:3_1_1"><i><font color="Green">E3</font></i></a></i>;<br/><a NAME="E5:3_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E4:3_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E7:3_1_1_2"/>
<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tsp_2.html#E3:3_1_1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="tsp_2.html#E4:3_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tex_4.html#T29">TEX_4:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tsp_1.html#V1">T_0</a>
 <b>by </b><i><a class="ref" href="tsp_2.html#D2" target="_self">Def2</a></i>;<br/>


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