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<b>let </b><font color="Maroon">X</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:24"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><br/> for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><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>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> ;<br/><a NAME="E1:24_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:24_1_1"/><i><font color="Green">E46</font></i>: 
 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> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>
<a NAME="E5:24_1_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E2:24_1"/>
 <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font> <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="zfmisc_1.html#T79">ZFMISC_1:79</a></i>;<br/><a NAME="E3:24_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">B</font> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:24_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:24_1_2"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  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> &amp; <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">a</font></span><a href="tarski.html#K1">}</a></span> ) )
 ;<br/><b>hence </b><a NAME="E3:24_1_2"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><a NAME="E6:24_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><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="E1:24_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_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="E1:24_1_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">B0</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>;<br/><a NAME="E1:24_1_3_1_1"/>
<font color="Maroon">B0</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="tdlat_3.html#E1:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E2:24_1_3_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B0</font> &amp; <font color="Maroon">B0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="txt" href="tdlat_3.html#E1:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_3_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:24_1_3"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E8:24_1"/><b>then </b>
(  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K1">bool</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a>, <a class="ref" href="zfmisc_1.html#T99">ZFMISC_1:99</a></i>;<br/><a NAME="E9:24_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#T26">TOPS_2:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V1" target="_self">discrete</a>
 <b>by </b><i/>;<br/>


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