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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="oppcat_1.html#M1" target="_self">Contravariant_Functor</a> of <font color="Maroon">C</font> <a href="oppcat_1.html#K2" target="_self">opp</a> ,<font color="Maroon">B</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:43_1"/>
for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font> st <span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E1:43_1_1"/>
<span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:43_1_1"/>
 ex <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font> st <span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">d</font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:43_1"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E3:43_1"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E4:43_1"/><i><font color="Green">E25</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43"/>
 <a href="oppcat_1.html#K7" target="_self">/*</a> <font color="Maroon">S</font> is    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>


</div>
