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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">C</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><a NAME="E1:96"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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> )
 ;<br/>

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



<b>assume </b><a NAME="E2:96"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K10" target="_self">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#R1">=</a> <font color="Maroon">h</font> )
 ;<br/>

<a NAME="E3:96"/><i><font color="Green">E42</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> <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">A</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">B</font>,<font color="Maroon">C</font> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E7:96"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:96"/><i><font color="Green">E43</font></i>: 
 <a href="funct_6.html#K10" target="_self">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="txt" href="funct_6.html#E7:96"><i><font color="Green">E53</font></i></a>, <a class="txt" href="funct_6.html#E5:96"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">cf</font> =  <a href="funct_6.html#K10" target="_self">commute</a> <font color="Maroon">f</font>;<br/>
<a NAME="E5:96"/><i><font color="Green">E52</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" target="_self">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="E6:96"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="funct_6.html#E9:96"><i><font color="Green">E54</font></i></a>, <a class="txt" href="funct_6.html#E10:96"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:96"/><b>then </b><i><font color="Green">E53</font></i>: 
 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> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</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="txt" href="funct_6.html#E9:96"><i><font color="Green">E54</font></i></a>, <a class="txt" href="funct_6.html#E10:96"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:96"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" target="_self">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" target="_self">commute</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="funct_6.html#E9:96"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:96"/><b>then </b><i><font color="Green">E54</font></i>: 
 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> <font color="Maroon">h</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="txt" href="funct_6.html#E9:96"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">uf</font> =  <a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font>;<br/>
<a NAME="E10:96"/><i><font color="Green">E55</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#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> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="txt" href="funct_6.html#E9:96"><i><font color="Green">E54</font></i></a>, <a class="txt" href="funct_6.html#E10:96"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>;<br/>
<b>thus </b><a NAME="E11:96"/>
( <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E9:96"><i><font color="Green">E54</font></i></a>, , , , <a class="ref" href="funct_5.html#T29">FUNCT_5:29</a></i>;<br/>


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