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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1"/>
( <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F<sub>1</sub></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="31_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> ;<br/><b>assume </b><a NAME="E2:31_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:31_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 ;<br/><a NAME="E4:31_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><b>hence </b><a NAME="E5:31_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="funct_7.html#E1:31_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:31_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_2"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>1</sub></font> = <font color="Maroon">F<sub>1</sub></font>() as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>);<br/><b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/><a NAME="E3:31_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E4:31_1_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="funct_7.html#E1:31"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/><a NAME="E5:31_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ,<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : verum </span>}</span> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D6">TARSKI:def 6</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:31_1_1_2_1_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:31_1_1_2_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>)
 <b>by </b><i><a class="txt" href="funct_7.html#E3:31_1_1_2_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E7:31_1_1_2_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="funct_7.html#E3:31_1_1_2_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="funct_7.html#E5:31_1_1_2_1_1"><i><font color="Green">E14</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_1_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:31_1_1_2_1_2"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>6</sub></font>) as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E5:31_1_1_2_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="funct_7.html#E3:31_1_1_2_1_2"><i><font color="Green">E13</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:31_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E5:31_1_1_2_1"/><i><font color="Green">E13</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>9</sub></font>),<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<b>assume </b><a NAME="E6:31_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E8:31_1_1_2_1"/><i><font color="Green">E14</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>10</sub></font>),<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E9:31_1_1_2_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>9</sub></font>) &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="funct_7.html#E5:31_1_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:31_1_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>10</sub></font>) &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="funct_7.html#E8:31_1_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:31_1_1_2_1"/>
( (  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; (  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="funct_7.html#E2:31"><i><font color="Green">E11</font></i></a>, <a class="txt" href="funct_7.html#E9:31_1_1_2_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:31_1_1_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="funct_7.html#E4:31_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/><a NAME="E7:31_1_1_2"/><i><font color="Green">E14</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="funct_7.html#E6:31_1_1_2"><i><font color="Green">E13</font></i></a>);<br/></i><a NAME="E8:31_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_1_1_2_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:31_1_1_2_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E3:31_1_1_2_2_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/>
<a NAME="E4:31_1_1_2_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="funct_7.html#E3:31_1_1_2_2_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1_2_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span> 
 ;<br/>


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

<b>let </b><font color="Maroon">c<sub>4</sub></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="E2:31_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>()<b> such that </b><br/><a NAME="E4:31_1_1_2_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  )
 ;<br/>
<a NAME="E5:31_1_1_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="funct_7.html#E4:31_1_1_2_2"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:31_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 <b>by </b><i><a class="txt" href="funct_7.html#E4:31_1_1_2_2"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:31_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="funct_7.html#E7:31_1_1_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
