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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font> holds   <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Olive" title="b1">A</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>



<a NAME="E1:52"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tex_4.html#T34" target="_self" title="TEX_4:th.34">Th34</a></i>;<br/>
<a NAME="E2:52"/>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<b>assume </b><a NAME="E1:52_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">C</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E4:52_1"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E6:52_1"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon" title="c4">C</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="c5">a</font>
 <b>and </b><br/><a NAME="E7:52_1"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E4:52_1"><i><font color="Green" title="E37">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:52_1"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D</font>
 <b>and </b><br/><a NAME="E10:52_1"/><i><font color="Green" title="E41">A7</font></i>: 
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Olive" title="b1">b</font></span>)</span> where <font color="Olive" title="b1">b</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font> : <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="tex_4.html#E7:52_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E12:52_1"/><i><font color="Green" title="E42">A8</font></i>: 
<font color="Maroon" title="c6">D</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="c7">b</font>
 <b>and </b><br/><a NAME="E13:52_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E10:52_1"><i><font color="Green" title="E41">A7</font></i></a></i>;<br/>
<a NAME="E14:52_1"/>
 <a href="tex_4.html#K2" title="TEX_4:func.2">MaxADSet</a> <font color="Maroon" title="c5">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="c7">b</font>
 
<b>by </b><i><a class="txt" href="tex_4.html#E9:52_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="tex_4.html#E12:52_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="tex_4.html#T23" target="_self" title="TEX_4:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E15:52_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="tex_4.html#E3:52_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="tex_4.html#E6:52_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="tex_4.html#E10:52_1"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="tex_4.html#E12:52_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:52"/>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="tex_4.html#E1:52"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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