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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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_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="32_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1"/><i><font color="Green">E31</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="32_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> ;<br/><b>assume </b><a NAME="E2:32_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</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="E3:32_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 ;<br/><a NAME="E4:32_1_1_1_1"/><b>then </b>
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><b>hence </b><a NAME="E5:32_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i>, <a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2"><b>suppose </b></a><a NAME="E1:32_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">A</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>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> ;<br/><a NAME="E3:32_1_1_2"/>
<font color="Maroon">A</font>,<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>),<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> where <font color="Olive">d</font> 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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1_1_2_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">d</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="E3:32_1_1_2_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>)
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">d</font> = <font color="Maroon">d</font>;<br/><b>thus </b><a NAME="E5:32_1_1_2_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E6:32_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:32_1_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">d</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="E3:32_1_1_2_1_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d</font>) as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E5:32_1_1_2_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <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> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E3:32_1_1_2_1"/>
<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> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d1</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="E5:32_1_1_2_1"/><i><font color="Green">E46</font></i>: 
<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#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d1</font>),<font color="Maroon">d1</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<b>assume </b><a NAME="E6:32_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d2</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="E8:32_1_1_2_1"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">u</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>3</sub></font>(<font color="Maroon">d2</font>),<font color="Maroon">d2</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E9:32_1_1_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d1</font>) &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d1</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:32_1_1_2_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">d2</font>) &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d2</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:32_1_1_2_1"/>
( (  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> ) &amp; (  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:32_1_1_2"/>
<font color="Maroon">F<sub>1</sub></font>(),<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">d</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  <a href="tarski.html#R2">are_equipotent</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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