<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V3">almost_discrete</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:112"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A</font> is <a href="tex_2.html#V3" target="_self">discrete</a>
 ;<br/>

<b>set </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> </span>}</span> ;<br/>
<a NAME="E2:112"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:112_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:112_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:112_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><b>thus </b><a NAME="E5:112_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:112_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">d</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> where <font color="Olive">d</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E5:112"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:112_2"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:112_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:112_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<a NAME="E5:112_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E6:112_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">S</font>,<font color="Maroon">d</font>]
 <b>by </b><i><a class="ref" href="tex_2.html#T18" target="_self">Th18</a>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:112"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">S</font>]
 <b>from </b><i><a class="ref" href="tex_2.html#S1" target="_self">TEX_2:sch 1</a>(<a class="ref" href="tex_2.html#T17" target="_self">Th17</a>);<br/></i>
<b>set </b><font color="Maroon">M</font> = <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>;<br/>
<a NAME="E8:112"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:112_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">S</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:112_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:112_3_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:112_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>thus </b><a NAME="E6:112_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_2.html#T18" target="_self">Th18</a>, <a class="ref" href="tex_2.html#D4" target="_self">Def4</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:112_3"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:112"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<a NAME="E10:112"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
;<br/>
<a NAME="E11:112"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T23">TDLAT_3:23</a></i>;<br/>
<a NAME="E12:112"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E13:112"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tex_2.html#D4" target="_self">Def4</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<a NAME="E14:112"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E15:112"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E16:112"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tex_2.html#D4" target="_self">Def4</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<a NAME="E17:112"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>thus </b><a NAME="E18:112"/>
 ex <font color="Olive">M</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">M</font> &amp; <font color="Olive">M</font> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E1:112_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>

<b>thus </b><a NAME="E2:112_4"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:112_4_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:112_4_1_1"/><i><font color="Green">E101</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_4_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:112_4_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="tex_2.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_4_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:112_4_1_1_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:112_4_1_1_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:112_4_1_1_1_1_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , </i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E4:112_4_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_2.html#T23" target="_self">Th23</a>, </i>;<br/><a NAME="E5:112_4_1_1_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:112_4_1_1_1_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/><a NAME="E7:112_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_2.html#E25"><i><font color="Green">Lemma58</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><a NAME="E8:112_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E9:112_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/><b>hence </b><a NAME="E10:112_4_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_2.html#E9:112"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:112_4_1_1_1_1_1"/>
 ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="112_4_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:112_4_1_1_1_1_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:112_4_1_1_1_1_2"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><a NAME="E2:112_4_1_1_1_1_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E3:112_4_1_1_1_1_2_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/><b>hence </b><a NAME="E4:112_4_1_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="tex_2.html#E9:112"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E5:112_4_1_1_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_2.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E6:112_4_1_1_1_1_2_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> )
 <b>by </b><i><a class="txt" href="tex_2.html#E9:112"><i><font color="Green">E64</font></i></a>, <a class="txt" href="tex_2.html#E12:112"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E7:112_4_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E8:112_4_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_4_1_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:112_4_1_1_1_1_2_1_1_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:112_4_1_1_1_1_2_1_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:112_4_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E5:112_4_1_1_1_1_2_1_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:112_4_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E7:112_4_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:112_4_1_1_1_1_2_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tex_2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E9:112_4_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tex_2.html#T25" target="_self">Th25</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">C</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:112_4_1_1_1_1_2_1_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E12:112_4_1_1_1_1_2_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E13:112_4_1_1_1_1_2_1_1_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">w</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E16:112_4_1_1_1_1_2_1_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E17:112_4_1_1_1_1_2_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E18:112_4_1_1_1_1_2_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_2.html#T18" target="_self">Th18</a>, , , </i>;<br/><a NAME="E19:112_4_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">w</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tex_2.html#D6" target="_self">Def6</a>, </i>;<br/><b>hence </b><a NAME="E20:112_4_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:112_4_1_1_1_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:112_4_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="tex_2.html#E14:112"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:112_4_1_1_1_1_2"/>
 ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:112_4_1_1"/>
 ex <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:112_4_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">F</font></span>)</span> is <a href="tex_2.html#V4" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="txt" href="tex_2.html#E8:112"><i><font color="Green">E50</font></i></a></i>;<br/>


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


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


</div>
