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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">f</font> be  <a href="functor0.html#V1" target="_self">Covariant</a> <a href="functor0.html#V2" target="_self">Contravariant</a> <a href="functor0.html#NM1">bifunction</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>consider </b><font color="Maroon">g1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E2:23"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g1</font>,<font color="Maroon">g1</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g2</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:23"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</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">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>
<b>take </b><font color="Maroon">b</font> = <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/>

<a NAME="E6:23"/><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.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>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><a NAME="E4:23_1"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:23_1"/><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:23_1"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/><a NAME="E7:23_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:23_1"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">a1</font>
<b>by </b><i>, <a class="txt" href="functor0.html#E4:23"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a1</font>,<font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="functor0.html#E6:23"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/><a NAME="E10:23_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E11:23_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">a2</font>
<b>by </b><i>, <a class="txt" href="functor0.html#E4:23"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">a</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="functor0.html#E6:23"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/><a NAME="E13:23_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E14:23_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> = 
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">g2</font>,<font color="Maroon">g2</font></span><a href="funct_3.html#K16">:]</a></span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">a1</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="functor0.html#E6:23"><i><font color="Green">E26</font></i></a>, , , <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:23"/>
<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>
 <b>by </b><i><a class="txt" href="functor0.html#E2:23"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>


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