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

<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">infinite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x0</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">T</font> =  <a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font>;<br/>
<a NAME="E1:51"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">SX</font> =  <a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">FX</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">F</font> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">F</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/>
<a NAME="E2:51"/><i><font color="Green">E31</font></i>: 
(  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/>
<a NAME="E3:51"/><i><font color="Green">E33</font></i>: 
1 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T13">CARD_4:13</a></i>;<br/>
<a NAME="E4:51"/><b>then </b>
<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">X</font></span>)</span> <a href="card_2.html#K1">+`</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="card_4.html#T34">CARD_4:34</a></i>;<br/>
<a NAME="E5:51"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="topgen_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="card_4.html#T76">CARD_4:76</a></i>;<br/>
<a NAME="E6:51"/><i><font color="Green">E37</font></i>: 
 <a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <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> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T46">EQREL_1:46</a></i>;<br/>
<b>let </b><font color="Maroon">B</font> be    <a href="cantor_1.html#M1">Basis</a> of  <a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font>;<br/>

<a NAME="E7:51"/>
<span class="p1">(<span class="default"><a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:51_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:51_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K11">SmallestPartition</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:51_1"/><b>then </b>
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x0</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E6:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span><b> such that </b><br/><a NAME="E5:51_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</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; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x0</font> )
 <b>by </b><i><a class="txt" href="topgen_2.html#E3:51"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:51_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topgen_2.html#E3:51"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">U</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topgen_2.html#K5" target="_self">DiscrWithInfin</a> <font color="Maroon">X</font>,<font color="Maroon">x0</font></span>)</span><b> such that </b><br/><a NAME="E8:51_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<a NAME="E9:51_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:51_1_1"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="txt" href="topgen_2.html#E5:51_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:51_1_1"/>
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E5:51_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:51_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E5:51_1"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:51"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="topgen_2.html#E5:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>


</div>
