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<span class="kw">thus </span><a NAME="E1:45_1_1"/>
( <font color="Maroon" title="c2">A</font> is  <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a>  implies  ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</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> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</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> ) implies <font color="Maroon" title="c2">A</font> is  <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</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/>

<a NAME="E2:45_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tex_4.html#T17" target="_self" title="TEX_4:th.17">Th17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:45_1_1_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</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="c4">x</font> )</span><br/>

<span class="kw">thus </span><a NAME="E6:45_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a></span>; <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="c2">A</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="c4">x</font></span><br/>

<a NAME="E7:45_1_1_1"/>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c4">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="E8:45_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c4">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/>
<a NAME="E9:45_1_1_1"/>
<font color="Maroon" title="c2">A</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="txt" href="#E1:45_1_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="tex_4.html#D7" target="_self" title="TEX_4:def.7">Def7</a></span>;<br/>
<a NAME="E10:45_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</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="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="tex_4.html#T21" target="_self" title="TEX_4:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:45_1_1_1"/>
<font color="Maroon" title="c2">A</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="c4">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E8:45_1_1_1"><span class="lab"><font color="Green" title="E29">A3</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="E2:45_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</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> )
 ; <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="c2">A</font> is  <a href="tex_4.html#V3" title="TEX_4:attr.3">maximal_anti-discrete</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:45_1_1"/>
<font color="Maroon" title="c2">A</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>; <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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