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<span class="kw">let </span><font color="Maroon" title="c1">Y</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> holds <br/> ( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b1">x</font> iff  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b1">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> iff  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> )</span><br/>



<a NAME="E1:39"/>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> is  <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T22" target="_self" title="TEX_4:th.22">Th22</a></span>;<br/>
<a NAME="E2:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> is  <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></span>;<br/>
<a NAME="E3:39"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> is  <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T22" target="_self" title="TEX_4:th.22">Th22</a></span>;<br/>
<a NAME="E4:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> is  <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:39"/>
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> implies  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E2:39_1"/><span class="kw">then </span>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> : ( <font color="Olive" title="b1">A</font> is  <a href="tex_4.html#V1" title="TEX_4:attr.1">anti-discrete</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>;<br/>
<a NAME="E3:39_1"/><span class="kw">then </span>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:39_1"/>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E6:39"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E7:39"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T14" target="_self" title="TEX_4:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:39"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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