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<b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Y</font>;<br/>



<a NAME="E1:37"/><i><font color="Green">E13</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:37"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:37"/>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font> is <a href="tex_4.html#V3" target="_self">maximal_anti-discrete</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:37"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font> implies  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> : ( <font color="Olive">A</font> is <a href="tex_4.html#V1" target="_self">anti-discrete</a> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tex_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E4:37_1"/>
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>assume </b><a NAME="E6:37"/><i><font color="Green">E22</font></i>: 
 <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E7:37"/>
<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#K2" target="_self">MaxADSet</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E4:37"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:37"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K2" target="_self">MaxADSet</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


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