<?xml version="1.0"?>
<div class="add">

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



<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a>
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2" target="_self">one-to-one</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:61"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:61_1"/>
( <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"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1_1"/>
<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"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E4:61_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:61_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</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">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E6:61_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:61_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E9:61_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">y</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>, </i>;<br/>
<a NAME="E10:61_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:61_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:61_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:61_1"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:61_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<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="E6:61_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E7:61_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:61_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E9:61_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">z</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>, </i>;<br/>
<a NAME="E10:61_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">z</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>, </i>;<br/>
<a NAME="E11:61_1"/><b>then </b>
<font color="Maroon">z</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>, </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:61_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E14:61_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<a NAME="E15:61_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , , , </i>;<br/>
<b>hence </b><a NAME="E16:61_1"/>
<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"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:61"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E6:61"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><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>thus </b><a NAME="E1:61_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E2:61_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E3:61_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:61_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:61"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E8:61"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_3"><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="E1:61_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:61_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</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/>;<br/>
<a NAME="E3:61_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:61_3"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:61_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1" target="_self">.</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K6">id</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:61"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:61"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K2" target="_self">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K2" target="_self">"</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
