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

<b>consider </b><font color="Maroon">M</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:122"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E3:122"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">X</font>, <font color="Olive">Y</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">M</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:122"/><i><font color="Green">E32</font></i>: 
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">M</font> holds <br/> ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> ) )
 <b>and </b><br/><a NAME="E5:122"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> or <font color="Olive">X</font>,<font color="Maroon">M</font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="tarski.html#T9">TARSKI:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E6:122"/>
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E7:122"/>
for <font color="Olive">X</font>, <font color="Olive">Y</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">M</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:122"/>
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">M</font> holds <br/> <a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1"><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><a NAME="E1:122_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:122_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:122_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="122_1_1"><b>now </b></a><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="E1:122_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:122_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:122_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:122_1"/><b>then </b>
 <a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:122_1"/>
 <a href="zfmisc_1.html#K1" target="_self">bool</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E9:122"/>
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> or <font color="Olive">X</font>,<font color="Maroon">M</font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/>


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