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

<b>consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:109_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:109_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:109_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:109_1"/><i><font color="Green">E46</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/>
<a NAME="E8:109_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_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>thus </b><a NAME="E1:109_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<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>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E10:109_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
;<br/>
<a NAME="E11:109_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E12:109_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:109_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) holds <br/> ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2"><b>proof </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><b>that </b><br/><a NAME="E1:109_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:109_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp;  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:109_1_2_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_2_1_1"><b>suppose </b></a><a NAME="E1:109_1_2_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>thus </b><a NAME="E3:109_1_2_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="109_1_2_1_2"><b>suppose </b></a><a NAME="E1:109_1_2_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x0</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E3:109_1_2_1_2"/>
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a></i>;<br/><b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x0</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="abcmiz_0.html#D9" target="_self">Def9</a></i>;<br/><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>  =  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">a<sub>1</sub></font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:109_1_2_1_2"/><i><font color="Green">E111</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</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">X</font> holds <br/><font color="Maroon">f</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><b>set </b><font color="Maroon">fx0</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x0</font> ) </span>}</span> ;<br/><a NAME="E7:109_1_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x0</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_2_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="E1:109_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x0</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_1_2_1"/><b>then </b>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x0</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">fx0</font> = <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x0</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> );<br/><a NAME="E9:109_1_2_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">fx0</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a>, </i>;<br/><a NAME="E10:109_1_2_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E11:109_1_2_1_2"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">k</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:109_1_2_1_2_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>given </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><a NAME="E2:109_1_2_1_2_2"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a>, <a class="ref" href="abcmiz_0.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>
<a NAME="E5:109_1_2_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E7:109_1_2_1_2_2"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> )
 ;<br/>
<b>set </b><font color="Maroon">fx0</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> ) </span>}</span> ;<br/>
<a NAME="E8:109_1_2_1_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_2_2_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="E1:109_1_2_1_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_1_2_2_1"/><b>then </b>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_1_2_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fx0</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> =  <a href="card_1.html#K4">card</a> <font color="Maroon">fx0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon">fx</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> ) </span>}</span> ;<br/>
<a NAME="E11:109_1_2_1_2_2"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_2_2_2"><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="E1:109_1_2_1_2_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:109_1_2_1_2_2_2"/><b>then </b>
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> )
 
;<br/>
<b>hence </b><a NAME="E3:109_1_2_1_2_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">fx</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E13:109_1_2_1_2_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">fx</font>
 
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#T10" target="_self">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<a NAME="E14:109_1_2_1_2_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">fx0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">fx</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2_1_2_2_3"><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="E1:109_1_2_1_2_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx0</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E3:109_1_2_1_2_2_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> )
 ;<br/>
<a NAME="E4:109_1_2_1_2_2_3"/>
<font color="Maroon">b</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a>, , <a class="ref" href="xboole_1.html#T56">XBOOLE_1:56</a></i>;<br/>
<b>hence </b><a NAME="E5:109_1_2_1_2_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E15:109_1_2_1_2_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">fx0</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">fx</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E16:109_1_2_1_2_2"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/>
<a NAME="E17:109_1_2_1_2_2"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  holds <br/>(  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> or  not <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Olive">a</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> )
 
;<br/>
<a NAME="E18:109_1_2_1_2_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">fx0</font> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E19:109_1_2_1_2_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">fx</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">fx0</font>
 
<b>by </b><i>, <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/>
<b>hence </b><a NAME="E20:109_1_2_1_2_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>

<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E21:109_1_2_1_2_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:109_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>from </b><i><a class="ref" href="nat_1.html#S7">NAT_1:sch 7</a>(, );<br/></i><b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E14:109_1_2_1_2"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/><a NAME="E15:109_1_2_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a></i>;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>;<br/><b>assume </b><a NAME="E16:109_1_2_1_2"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E17:109_1_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a>, <a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E18:109_1_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E19:109_1_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">z</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">y</font> ) </span>}</span>  or <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D12" target="_self">Def12</a></i>;<br/><b>hence </b><a NAME="E20:109_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">x</font>]
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D11" target="_self">Def11</a>, <a class="ref" href="abcmiz_0.html#T10" target="_self">Th10</a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E15:109_1"/><i><font color="Green">E179</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font>  st <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Maroon">x</font>]
 <b>from </b><i><a class="ref" href="orders_1.html#S2">ORDERS_1:sch 2</a>(, , , );<br/></i>
<a NAME="E16:109_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E18:109_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/>

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

<b>assume </b><a NAME="E19:109_1"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">B</font>] )
 ;<br/>

<a NAME="E20:109_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:109_1"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:109_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E23:109_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D9" target="_self">Def9</a>, <a class="ref" href="abcmiz_0.html#T8" target="_self">Th8</a></i>;<br/>


</div>
