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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>;<br/><b>let </b><font color="Maroon">F</font> be  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a>
 ;<br/>

<a NAME="E2:55"/>
( <font color="Maroon">A</font> is    <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> is    <a href="altcat_2.html#M1">SubCatStr</a> of <font color="Maroon">B</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> <a href="pboole.html#R6">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A</font> &amp; the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> <a href="pboole.html#R6">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="altcat_2.html#T21">ALTCAT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E3:55"/>
( <font color="Maroon">A</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="altcat_4.html#T35">ALTCAT_4:35</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">F</font>
;<br/>

<b>thus </b><a NAME="E4:55"/>
( <font color="Maroon">F</font> is <a href="functor0.html#V21">bijective</a> &amp; ( for <font color="Olive">a'</font>, <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>  st <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> holds <br/><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> ) )
 <b>by </b><i><a class="txt" href="yellow20.html#E5:55"><i><font color="Green">E32</font></i></a></i>;<br/>

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



<b>assume </b><a NAME="E5:55"/><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">b</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E6:55"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</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>hence </b><a NAME="E7:55"/>
for <font color="Olive">f'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">c'</font><br/> for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>  st <font color="Olive">f'</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> holds <br/><font color="Maroon">F</font> <a href="functor0.html#K8">.</a> <font color="Olive">f'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f</font>
 <b>by </b><i>, <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>;<br/>


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