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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">B</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A</font>;<br/>



<b>set </b><font color="Maroon">F</font> =  <a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span>;<br/>
<a NAME="E1:64"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A</font>,<font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a>  <a href="yellow18.html#R2">are_opposite</a>  &amp; <font color="Maroon">B</font>,<font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a>  <a href="yellow18.html#R2">are_opposite</a>  )
 
<b>by </b><i><a class="ref" href="yellow18.html#D4">YELLOW18:def 4</a></i>;<br/>
<b>thus </b><a NAME="E2:64"/>
( <font color="Maroon">B</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a>  is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a>  )
 <b>by </b><i><a class="ref" href="yellow20.html#T3" target="_self">Th3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self">YELLOW20:def 5</a><br/>

<b>take </b><font color="Maroon">G</font> =  <a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="yellow18.html#K1">opp</a> </span>)</span>;<br/>

<b>thus </b><a NAME="E3:64"/>
<font color="Maroon">G</font> is <a href="functor0.html#V21">bijective</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 ;<br/><b>hence </b><font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D5">YELLOW18:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a1</font>
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow18.html#D5">YELLOW18:def 5</a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">c1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E5:64"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">c1</font>;<br/>



<b>assume </b><a NAME="E6:64"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 ;<br/>

<a NAME="E7:64"/><i><font color="Green">E34</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">c1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E8:64"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">A</font>,<span class="p4">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, , <a class="ref" href="yellow18.html#D5">YELLOW18:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">A</font>,<span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font>
<b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="txt" href="yellow20.html#E8:64"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow18.html#D5">YELLOW18:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2">dualizing-func</a> <font color="Maroon">A</font>,<span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="yellow18.html#K1">opp</a> </span>)</span></span>)</span>,<font color="Maroon">b1</font>,<font color="Maroon">c1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f1</font>
<b>by </b><i><a class="txt" href="yellow20.html#E8:64"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow20.html#D1" target="_self">Def1</a>, <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
;<br/>


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