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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:66_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">F</font> is <a href="functor0.html#V14" target="_self">comp-reversing</a>
 ;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:66_1_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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> 
 <b>and </b><br/><a NAME="E3:66_1_1_1"/><i><font color="Green">E26</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</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> 
 ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/><b>consider </b><font color="Maroon">f'</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</font></span>)</span>, <font color="Maroon">g'</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E5:66_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:66_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E7:66_1_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g'</font>
 <b>by </b><i><a class="ref" href="functor0.html#T4" target="_self">Th4</a>, <a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, </i>;<br/><a NAME="E8:66_1_1_1"/><i><font color="Green">E34</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">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</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="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, </i>;<br/><a NAME="E9:66_1_1_1"/>
<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" target="_self">.</a> <font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</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="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, </i>;<br/><a NAME="E10:66_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="functor0.html#E4:66_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="functor0.html#T5" target="_self">Th5</a>, <a class="ref" href="functor0.html#T6" target="_self">Th6</a>, <a class="txt" href="functor0.html#E7:66_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E11:66_1_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</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> 
 <b>by </b><i><a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/><a NAME="E12:66_1_1_1"/><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" target="_self">.</a> <font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</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/>;<br/><b>hence </b><a NAME="E13:66_1_1_1"/>
<font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">f</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="functor0.html#E7:66_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:66_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">o1</font>, <font color="Olive">o2</font>, <font color="Olive">o3</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>  st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o1</font>,<font color="Olive">o2</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; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o2</font>,<font color="Olive">o3</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>  holds <br/>for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o1</font>,<font color="Olive">o2</font><br/> for <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o2</font>,<font color="Olive">o3</font> holds  <font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Olive">f</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Olive">g</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D23" target="_self">FUNCTOR0:def 23</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:66_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</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> 
 <b>and </b><br/><a NAME="E4:66_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</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> 
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/>



<a NAME="E5:66_1_1"/><i><font color="Green">E63</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">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</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="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">f'</font> = <span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</font></span>)</span> <b>by </b><i><a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E7:66_1_1"/><i><font color="Green">E66</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">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</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="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">g'</font> = <span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font></span>)</span> <b>by </b><i><a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>take </b>
<font color="Maroon">f'</font>
;<br/>

<b>take </b>
<font color="Maroon">g'</font>
;<br/>

<b>thus </b><a NAME="E9:66_1_1"/>
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o2</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E10:66_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="functor0.html#E4:66_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="functor0.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E11:66_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</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> 
 
<b>by </b><i><a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E12:66_1_1"/><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" target="_self">.</a> <font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</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/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="functor0.html#K7" target="_self">Morph-Map</a> <font color="Maroon">F</font>,<font color="Maroon">o1</font>,<font color="Maroon">o3</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">F</font> <a href="functor0.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T6" target="_self">Th6</a>, </i>
<br/>.= 
<font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="functor0.html#T4" target="_self">Th4</a>, <a class="txt" href="functor0.html#E2:66_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="functor0.html#E3:66_1_1"><i><font color="Green">E46</font></i></a>, </i>
;<br/><br/>


</div>
