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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><a NAME="E1:41"/>
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> is <a href="functor0.html#V21">bijective</a>
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="functor0.html#R1">are_isomorphic</a> 
 ;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="functor0.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V15">covariant</a> )
 <b>by </b><i><a class="ref" href="functor0.html#D40">FUNCTOR0:def 40</a></i>;<br/><b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">A</font> <b>by </b><i/>;<br/><a NAME="E5:41_1"/>
( <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a> &amp; <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">F</font> is <a href="functor0.html#V16">contravariant</a> )
 <b>by </b><i>, , <a class="ref" href="functor1.html#T13">FUNCTOR1:13</a></i>;<br/><b>hence </b><a NAME="E6:41_1"/>
<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="functor0.html#R2">are_anti-isomorphic</a> 
 <b>by </b><i><a class="ref" href="functor0.html#D41">FUNCTOR0:def 41</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:41"/>
<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="functor0.html#R2">are_anti-isomorphic</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being    <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a> &amp; <font color="Maroon">F</font> is <a href="functor0.html#V16">contravariant</a> )
 <b>by </b><i><a class="ref" href="functor0.html#D41">FUNCTOR0:def 41</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E8:41"/>
( <font color="Maroon">F</font> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> is <a href="functor0.html#V21">bijective</a> &amp; <font color="Maroon">F</font> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> is <a href="functor0.html#V15">covariant</a> )
 
<b>by </b><i>, , <a class="ref" href="functor1.html#T13">FUNCTOR1:13</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
<font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="functor0.html#R1">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="functor0.html#D40">FUNCTOR0:def 40</a></i>;<br/>


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