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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</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/>





<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>set </b><font color="Maroon">cf</font> =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font>;<br/>
<a NAME="E3:3"/>
 <a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T85">FUNCT_6:85</a></i>;<br/>
<a NAME="E4:3"/><b>then </b><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">g</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E5:3"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:3"/><b>then </b>
 ex <font color="Olive">t</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">t</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:3"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 ;<br/>


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