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

<b>thus </b><a NAME="E1:89_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="functor0.html#V12" target="_self">id-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D21" target="_self">FUNCTOR0:def 21</a><br/>

<a NAME="E1:89_1_1_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></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/>
<b>hence </b><span class="p1">(<span class="default"><a href="functor0.html#K5" target="_self">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="functor0.html#T31" target="_self">Th31</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#T30" target="_self">Th30</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:89_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="functor0.html#V13" target="_self">comp-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></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><b>that </b><br/><a NAME="E1:89_1_1_2"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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="E2:89_1_1_2"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>



<a NAME="E3:89_1_1_2"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></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#E1:89_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="functor0.html#E2:89_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E4:89_1_1_2"/><i><font color="Green">E65</font></i>: 
( <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K3" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="functor0.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E5:89_1_1_2"/>
( <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="functor0.html#E1:89_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="functor0.html#E2:89_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="functor0.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E6:89_1_1_2"/>
<span class="p1">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="functor0.html#K6" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="functor0.html#E3:89_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="functor0.html#E4:89_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="functor0.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:89_1_1"/>
 <a href="functor0.html#K11" target="_self">id</a> <font color="Maroon">c<sub>1</sub></font> is <a href="functor0.html#V13" target="_self">comp-preserving</a>
 ;<br/>


</div>
