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<div class="add">

<b>let </b><font color="Maroon" title="c3">r</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font>;<br/>

<a NAME="E1:48_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:48_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="tsp_2.html#K3" title="TSP_2:func.3">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font> implies for <font color="Olive" title="b1">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> holds <br/>for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="tsp_2.html#K3" title="TSP_2:func.3">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">N</font> = <font color="Maroon" title="c4">M</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>assume </b><a NAME="E3:48_1_1_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ;<br/>

<a NAME="E4:48_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">N</font> is <a href="tsp_2.html#V1" title="TSP_2:attr.1">maximal_T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#T11" target="_self" title="TSP_2:th.11">Th11</a></i>;<br/>
<a NAME="E5:48_1_1_1"/><b>then </b><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c5">N</font> is <a href="tsp_1.html#V1" title="TSP_1:attr.1">T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#D4" target="_self" title="TSP_2:def.4">Def4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E6:48_1_1_1"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1_1_1"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c7">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:48_1_1_1_1_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:48_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:48_1_1_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="tsp_2.html#D2" target="_self" title="TSP_2:def.2">Def2</a></i>;<br/>
<a NAME="E5:48_1_1_1_1_1"/>
<font color="Maroon" title="c8">c</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="c6">A</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1_1_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:48_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">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">X</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tex_4.html#D11" title="TEX_4:def.11">TEX_4:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:48_1_1_1_1_1"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">D</font>
 <b>and </b><br/><a NAME="E9:48_1_1_1_1_1"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c9">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#K4" title="TEX_4:func.4">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">X</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="c10">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E11:48_1_1_1_1_1"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c10">a</font>
 <b>and </b><br/><a NAME="E12:48_1_1_1_1_1"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E9:48_1_1_1_1_1"><i><font color="Green" title="E42">A9</font></i></a></i>;<br/>
<a NAME="E13:48_1_1_1_1_1"/>
 <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c10">a</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E8:48_1_1_1_1_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="tsp_2.html#E11:48_1_1_1_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="tex_4.html#T23" title="TEX_4:th.23">TEX_4:23</a></i>;<br/>
<a NAME="E14:48_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="tsp_2.html#E3:48_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="tsp_2.html#E4:48_1_1_1_1_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="tsp_2.html#D10" target="_self" title="TSP_2:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E15:48_1_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E12:48_1_1_1_1_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_1_1_1_1"/>
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:48_1_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:48_1_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_1_1_2_1"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">X0</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:48_1_1_1_2_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E5:48_1_1_1_2_1"/><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 <b>and </b><br/><a NAME="E6:48_1_1_1_2_1"/><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1_2_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a</font> = <font color="Maroon" title="c9">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="tsp_2.html#E4:48_1_1_1_2_1"><i><font color="Green" title="E40">A13</font></i></a></i>;<br/>
<a NAME="E8:48_1_1_1_2_1"/><i><font color="Green" title="E43">A16</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="tsp_2.html#E3:48_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="tsp_2.html#E6:48_1_1_1_2_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="tsp_2.html#D10" target="_self" title="TSP_2:def.10">Def10</a></i>;<br/>
<a NAME="E9:48_1_1_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">a</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> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:48_1_1_1_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:48_1_1_1_2_1"/><b>then </b>
 <a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">a</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#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T33" title="TEX_4:th.33">TEX_4:33</a></i>;<br/>
<a NAME="E11:48_1_1_1_2_1"/><b>then </b>
 <a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c10">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> <font color="Maroon" title="c6">A</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T30" title="TEX_4:th.30">TEX_4:30</a></i>;<br/>
<a NAME="E12:48_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<b>hence </b><a NAME="E13:48_1_1_1_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="tsp_2.html#E8:48_1_1_1_2_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_1_1_1_2"/>
<font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:48_1_1_1"/>
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">A</font>
 <b>by </b><i><a class="txt" href="tsp_2.html#E6:48_1_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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

<b>assume </b><a NAME="E3:48_1_1"/><i><font color="Green" title="E35">A17</font></i>: 
for <font color="Olive" title="b1">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> holds <br/>for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <font color="Olive" title="b2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">A</font>
 ;<br/>

<a NAME="E4:48_1_1"/>
for <font color="Olive" title="b1">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Olive" title="b1">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:48_1_1_2"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>

<a NAME="E2:48_1_1_2"/>
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K3" title="TEX_4:func.3">MaxADSet</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E3:48_1_1"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="tsp_2.html#E1:48_1_1_2"><i><font color="Green" title="E36">A18</font></i></a></i>;<br/>
<a NAME="E3:48_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">r</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T30" title="TEX_4:th.30">TEX_4:30</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1_1_2"/>
<font color="Maroon" title="c4">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4" title="TEX_4:func.4">MaxADSet</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="tsp_2.html#E1:48_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:48_1_1"/>
<font color="Maroon" title="c3">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="tsp_2.html#K3" title="TSP_2:func.3">Stone-retraction</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font>
 <b>by </b><i><a class="ref" href="tsp_2.html#D10" target="_self" title="TSP_2:def.10">Def10</a></i>;<br/>


</div>
