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<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:11"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#T45">FUNCT_4:45</a></i>;<br/>
<b>assume </b><a NAME="E4:11"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x11</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x12</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:11"/>
( <font color="Maroon">x11</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x12</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E7:11"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x11</font>,<font color="Maroon">x12</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>assume </b><a NAME="E8:11"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x21</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x22</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:11"/>
( <font color="Maroon">x21</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x22</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E11:11"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x21</font>,<font color="Maroon">x22</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>assume </b><a NAME="E12:11"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E13:11"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x12</font>,<font color="Maroon">x11</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_4.html#T43">FUNCT_4:43</a></i>;<br/>
<a NAME="E14:11"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x22</font>,<font color="Maroon">x21</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_4.html#T43">FUNCT_4:43</a></i>;<br/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x12</font>,<font color="Maroon">x11</font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T44">FUNCT_4:44</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x22</font>,<font color="Maroon">x21</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T44">FUNCT_4:44</a></i>
;<br/>
<a NAME="E16:11"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x12</font>,<font color="Maroon">x11</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">x22</font>,<font color="Maroon">x21</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E17:11"/><b>then </b>
( <font color="Maroon">x12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x22</font> &amp; <font color="Maroon">x21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x11</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E18:11"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/>


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