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<b>thus </b><a NAME="E1:35_1_1"/>
 <a href="functor0.html#K10">incl</a> <font color="Maroon">C</font> is <a href="functor0.html#V12">id-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:35_1_1_1"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="functor0.html#T28">FUNCTOR0:28</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="functor0.html#K5">Morph-Map</a> <span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">a</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="functor0.html#D16">FUNCTOR0:def 16</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font>
<b>by </b><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#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
;<br/>


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

<b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D24">FUNCTOR0:def 24</a><br/>

<b>assume </b><a NAME="E2:35_1_1"/><i><font color="Green">E32</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/>

<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="E3:35_1_1"/>
<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/>
<a NAME="E4:35_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o3</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</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">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , <a class="ref" href="functor0.html#T28">FUNCTOR0:28</a></i>;<br/>
<b>hence </b><a NAME="E5:35_1_1"/>
<span class="p1">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</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"><span class="p2">(<span class="default"><a href="functor0.html#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">g</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#K10">incl</a> <font color="Maroon">C</font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>;<br/>


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