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

<b>let </b><font color="Maroon">A</font> be  <a href="card_4.html#V1">uncountable</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">TS</font> =  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ;<br/>
<a NAME="E1:32"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</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="E1:32_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>
<a NAME="E4:32_1"/>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E5:32_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:32_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><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"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E3:32"/>
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>  ex <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Z</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>;<br/>

<b>set </b><font color="Maroon">Z</font> = <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>;<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <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> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> ;<br/>
<a NAME="E2:32_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:32_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">UZ</font> = <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">xAAx</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>;<br/>
<b>set </b><font color="Maroon">yAAy</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>;<br/>
<a NAME="E5:32_2"/>
<span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U</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="E1:32_2_1"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">O</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:32_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font> )
 ;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:32_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:32_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_2_1_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 </span><b>by </b><i>, , <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="32_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:32_2_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:32_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:32_2_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:32_2_1_1_1_1"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:32_2_1_1_1_1"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:32_2_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:32_2_1_1_1_2"/><b>then </b>
<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="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:32_2_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:32_2_1_1_1_2"/><b>then </b>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:32_2_1_1_1_2"/>
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:32_2_1"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span></span><a href="tarski.html#K1">.}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:32_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:32_2"/>
 ex <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">{<span class="default"> <font color="Olive">O</font> where <font color="Olive">O</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> : ( <font color="Olive">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Z</font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32"/>
<font color="Maroon">F</font> is <a href="pcomps_1.html#V1">locally_finite</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D2">PCOMPS_1:def 2</a></i>;<br/>

<a NAME="E5:32"/>
not <font color="Maroon">F</font> is <a href="nagata_1.html#V4" target="_self">sigma_discrete</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3"/>
<font color="Maroon">F</font> is <a href="nagata_1.html#V4" target="_self">sigma_discrete</a>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being  <a href="nagata_1.html#V2" target="_self">sigma_discrete</a> <a href="nagata_1.html#NM1">FamilySequence</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E3:32_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="nagata_1.html#K2" target="_self">Union</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:32_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">aAAa</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<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> <font color="Maroon">f</font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Maroon">f</font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> ) or ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">f</font> <a href="cantor_1.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a> ) );<br/>
<a NAME="E7:32_3"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/> ex <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">f</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1"/>
( <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_1"/>
<font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>] )
 ;<br/><b>hence </b><a NAME="E3:32_3_1_1_1_1"/>
 ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">A</font>] )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2"/>
not <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:32_3_1_1_1_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">A</font>] )
 <b>by </b><i>, , <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_2"/>
 ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">A</font>] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1"/>
 ex <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Olive">f</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">Df</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">F</font><b> such that </b><br/><a NAME="E9:32_3"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">Df</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E10:32_3"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">AD</font>, <font color="Olive">BD</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">AD</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">BD</font>] holds <br/><font color="Olive">AD</font> <a href="hidden.html#R1">=</a> <font color="Olive">BD</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">AD</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>, <font color="Maroon">BD</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F</font>;<br/>



<b>assume </b><a NAME="E1:32_3_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">AD</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">BD</font>] )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_2_1_1"/>
( <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_2_1_1_1"/>
<font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_2_1_1_1"/>
<font color="Maroon">AD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BD</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_2_1_1_2"/>
not <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:32_3_2_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">AD</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">AD</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">BD</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">BD</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">n</font> is <a href="nagata_1.html#V1" target="_self">discrete</a> )
 <b>by </b><i>, , , <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:32_3_2_1_1_2"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">AD</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E6:32_3_2_1_1_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">BD</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><a NAME="E7:32_3_2_1_1_2"/>
<font color="Maroon">AD</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">BD</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_2_1_1_2_1"/>
( <font color="Maroon">a</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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</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/>
<a NAME="E2:32_3_2_1_1_2_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:32_3_2_1_1_2_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">AD</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">BD</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_2_1_1_2_1"/>
<font color="Maroon">AD</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">BD</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:32_3_2_1_1_2"/>
<font color="Maroon">AD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BD</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_2"/>
<font color="Maroon">AD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">BD</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:32_3"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">cAAc</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="E1:32_3_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">cAAc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">cAAc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="nagata_1.html#K1" target="_self">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E4:32_3_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">cAAc</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">Df</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>] )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:32_3_3"/><b>then </b>
( <font color="Maroon">cAAc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Df</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">Df</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_3"/>
<font color="Maroon">cAAc</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:32_3"/><i><font color="Green">E117</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_4"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:32_3_4"/><i><font color="Green">E119</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E3:32_3_4"/><i><font color="Green">E120</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">AA</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="E1:32_3_4_1"/>
<font color="Maroon">AA</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">d</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_3_4_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> &amp; <font color="Maroon">AA</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/>
<a NAME="E5:32_3_4_1"/>
<font color="Maroon">AA</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:32_3_4_1"/>
<font color="Maroon">AA</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_4"/>
for <font color="Olive">a1</font>, <font color="Olive">a2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> &amp; <font color="Olive">a2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a2</font> holds <br/><font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_4_2"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> )
 ;<br/>

<a NAME="E2:32_3_4_2"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a2</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>assume </b><a NAME="E3:32_3_4_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a2</font>
 ;<br/>

<a NAME="E4:32_3_4_2"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a1</font></span><a href="tarski.html#K1">.}</a></span> &amp; <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E5:32_3_4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a1</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:32_3_4_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a2</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a2</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:32_3_4_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a2</font></span><a href="tarski.html#K1">.}</a></span>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> or <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a2</font></span><a href="tarski.html#K1">.}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:32_3_4_2"/><b>then </b>
<font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a2</font></span><a href="tarski.html#K1">.}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3_4_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_4"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">d</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_4"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3"/>
not  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>
<a NAME="E14:32_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a>, <a class="ref" href="card_1.html#T38">CARD_1:38</a></i>;<br/>
<a NAME="E15:32_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font> &amp;  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<a NAME="E16:32_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  &amp;  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#R2">c&lt;</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="card_2.html#T3">CARD_2:3</a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E17:32_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="xboole_0.html#R2">c&lt;</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T59">XBOOLE_1:59</a></i>;<br/>
<a NAME="E18:32_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E19:32_3"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<a NAME="E20:32_3"/><b>then </b>
<font color="Maroon">F</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">Df</font> <a href="relset_1.html#K10">.:</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4">CARD_2:4</a></i>;<br/>
<b>hence </b><a NAME="E21:32_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
not <font color="Maroon">F</font> is <a href="nagata_1.html#V4" target="_self">sigma_discrete</a>
 ;<br/>


</div>
