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

<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><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">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="functor0.html#NM1">bifunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>







<b>assume </b><a NAME="E1:21"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">A</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">b</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>thus </b><a NAME="E3:21"/>
<font color="Maroon">f</font> is <a href="functor0.html#V1" target="_self">Covariant</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D2" target="_self">FUNCTOR0:def 2</a><br/>

<b>thus </b><a NAME="E1:21_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i>, <a class="ref" href="altcat_2.html#T1">ALTCAT_2:1</a></i>;<br/>


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

<b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D3" target="_self">FUNCTOR0:def 3</a><br/>

<a NAME="E4:21"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="funct_2.html#R2">=</a>  <a href="functor0.html#K1" target="_self">~</a> <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:21"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K1" target="_self">~</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">g</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i>, <a class="ref" href="altcat_2.html#T1">ALTCAT_2:1</a></i>;<br/>


</div>
