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<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<a NAME="E1:5"/><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T30">FUNCT_5:30</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">b</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_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/><b>consider </b><font color="Maroon">a'</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">b'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:5_1"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a'</font>,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">g</font>;<br/><b>thus </b><a NAME="E6:5_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E3:5"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E4:5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:5"/>
<font color="Maroon">b</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_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>


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