<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">A2</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">A3</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">A1</font>,<font color="Maroon">A2</font>;<br/><b>let </b><font color="Maroon">G</font> be  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A2</font>,<font color="Maroon">A3</font>;<br/><b>let </b><font color="Maroon">B1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A1</font>;<br/><b>let </b><font color="Maroon">B2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A2</font>;<br/><b>let </b><font color="Maroon">B3</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A3</font>;<br/>











<b>assume </b><a NAME="E1:70"/>
( <font color="Maroon">B1</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A1</font> &amp; <font color="Maroon">B2</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self">YELLOW20:def 5</a><br/>

<b>given </b><font color="Maroon">F1</font> being  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B1</font>,<font color="Maroon">B2</font><b> such that </b><a NAME="E2:70"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">F1</font> is <a href="functor0.html#V21">bijective</a>
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B1</font><br/> for <font color="Olive">a1</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1</font> holds <br/><font color="Maroon">F1</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">a1</font>
 <b>and </b><br/><a NAME="E4:70"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B1</font><br/> for <font color="Olive">b1</font>, <font color="Olive">c1</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b</font>,<font color="Olive">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="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c1</font> holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b</font>,<font color="Olive">c</font><br/> for <font color="Olive">f1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b1</font>,<font color="Olive">c1</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/><font color="Maroon">F1</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#K7">Morph-Map</a> <font color="Maroon">F</font>,<font color="Olive">b1</font>,<font color="Olive">c1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f1</font>
 ;<br/>

<b>assume </b><a NAME="E5:70"/>
( <font color="Maroon">B2</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A2</font> &amp; <font color="Maroon">B3</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A3</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D5" target="_self">YELLOW20:def 5</a><br/>

<b>given </b><font color="Maroon">G1</font> being  <a href="functor0.html#V16">contravariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B2</font>,<font color="Maroon">B3</font><b> such that </b><a NAME="E6:70"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G1</font> is <a href="functor0.html#V21">bijective</a>
 <b>and </b><br/><a NAME="E7:70"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B2</font><br/> for <font color="Olive">a1</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A2</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1</font> holds <br/><font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <font color="Olive">a1</font>
 <b>and </b><br/><a NAME="E8:70"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B2</font><br/> for <font color="Olive">b1</font>, <font color="Olive">c1</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A2</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b</font>,<font color="Olive">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="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c1</font> holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b</font>,<font color="Olive">c</font><br/> for <font color="Olive">f1</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b1</font>,<font color="Olive">c1</font>  st <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> holds <br/><font color="Maroon">G1</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#K7">Morph-Map</a> <font color="Maroon">G</font>,<font color="Olive">b1</font>,<font color="Olive">c1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">f1</font>
 ;<br/>

<b>thus </b><a NAME="E9:70"/>
( <font color="Maroon">B1</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A1</font> &amp; <font color="Maroon">B3</font> is   <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">A3</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow20.html#D4" target="_self">YELLOW20:def 4</a><br/>

<b>take </b>
<font color="Maroon">G1</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F1</font>
;<br/>

<b>thus </b><a NAME="E10:70"/>
<font color="Maroon">G1</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F1</font> is <a href="functor0.html#V21">bijective</a>
 <b>by </b><i><a class="ref" href="yellow20.html#T4" target="_self">Th4</a>, <a class="txt" href="yellow20.html#E7:70"><i><font color="Green">E39</font></i></a>, <a class="ref" href="functor1.html#T13">FUNCTOR1:13</a></i>;<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">B1</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font>;<br/><b>assume </b><a NAME="E1:70_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:70_1"/><b>then </b>
<font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E3:70_1"/><b>then </b>
<font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow20.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</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">B1</font>, <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B1</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font>, <font color="Maroon">c1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font>;<br/>



<b>assume </b><a NAME="E12:70"/><i><font color="Green">E52</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/>



<a NAME="E13:70"/><i><font color="Green">E53</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</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>  &amp; <font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font> )
 
<b>by </b><i>, <a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, <a class="ref" href="functor0.html#D20">FUNCTOR0:def 20</a></i>;<br/>
<a NAME="E14:70"/><b>then </b><i><font color="Green">E54</font></i>: 
( <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> &amp; <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">F1</font> <a href="functor0.html#K8">.</a> <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"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</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="tarski.html#R1">c=</a> <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">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, <a class="ref" href="altcat_2.html#T32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E15:70"/><b>then </b><i><font color="Green">E55</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"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</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>  &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<b>assume </b><a NAME="E16:70"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font>
 ;<br/>

<b>then </b><font color="Maroon">F1</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">F</font>,<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="ref" href="yellow20.html#T5" target="_self">Th5</a>, <a class="ref" href="yellow20.html#T10" target="_self">Th10</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
;<br/>
<a NAME="E18:70"/><b>then </b>
<font color="Maroon">G1</font> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">G</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow20.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="functor0.html#K7">Morph-Map</a> <font color="Maroon">G</font>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">c1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3">.</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow20.html#T10" target="_self">Th10</a>, <a class="ref" href="functor3.html#T7">FUNCTOR3:7</a></i>
<br/>.= 
<font color="Maroon">G</font> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8">.</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="functor0.html#D17">FUNCTOR0:def 17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f1</font>
<b>by </b><i>, <a class="ref" href="functor3.html#T7">FUNCTOR3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="functor3.html#K2">*</a> <font color="Maroon">F</font></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="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>
;<br/><br/>


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