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

<b>set </b><font color="Maroon">F</font> = <font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span> as  <a href="functor0.html#V8" target="_self">feasible</a> <a href="functor0.html#V10" target="_self">Covariant</a>  <a href="functor0.html#L2" target="_self">FunctorStr</a> of <font color="Maroon">C1</font>,<font color="Maroon">C2</font> ;<br/>
<a NAME="E2:69_1_1"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">o</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="ref" href="altcat_2.html#D7">ALTCAT_2:def 7</a></i>;<br/>
<a NAME="E3:69_1_1"/>
<font color="Maroon">G</font> is <a href="functor0.html#V13" target="_self">comp-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<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#D24" target="_self">FUNCTOR0:def 24</a><br/>

<b>assume </b><a NAME="E1:69_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>  &amp; <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/>

<a NAME="E2:69_1_1_1"/><b>then </b><i><font color="Green">E26</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="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<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="E3:69_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">G</font> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font> &amp; <font color="Maroon">G</font> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, <a class="ref" href="functor0.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:69_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">G</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">G</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <font color="Maroon">G</font> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">G</font> <a href="functor0.html#K6" 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="altcat_1.html#K8">idm</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="functor0.html#T5" target="_self">Th5</a>, <a class="ref" href="functor0.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">g</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="functor0.html#T6" target="_self">Th6</a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:69_1_1"/>
<font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span> is <a href="functor0.html#V13" target="_self">comp-preserving</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D25" target="_self">FUNCTOR0:def 25</a><br/>

<b>assume </b><a NAME="E5:69_1_1"/><i><font color="Green">E33</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>  &amp; <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/>

<a NAME="E6:69_1_1"/><b>then </b><i><font color="Green">E34</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="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<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="E7:69_1_1"/><i><font color="Green">E36</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:69_1_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <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="altcat_1.html#K8">idm</a> <font color="Maroon">o</font>
<b>by </b><i><a class="ref" href="functor0.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="functor0.html#K9" target="_self">--&gt;</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="functor0.html#K8" target="_self">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="functor0.html#T6" target="_self">Th6</a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
;<br/>


</div>
