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

<b>let </b><font color="Maroon">UN</font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>set </b><font color="Maroon">C</font> =  <a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="modcat_1.html#M4" target="_self">Element</a> of  <a href="modcat_1.html#K4" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span> ;<br/>
<i><font color="Green">E33</font></i>:  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K8" target="_self">dom</a> <span class="p2">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D2">CAT_1:def 2</a></i>
<br/>.= 
 <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">g'</font>
<b>by </b><i/>
<br/>.= 
 <a href="mod_2.html#K2">dom</a> <font color="Maroon">g'</font>

;<br/>
<i><font color="Green">E37</font></i>:  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K9" target="_self">cod</a> <span class="p2">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f'</font>
<b>by </b><i><a class="ref" href="cat_1.html#D3">CAT_1:def 3</a></i>
<br/>.= 
 <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">f'</font>
<b>by </b><i/>
<br/>.= 
 <a href="mod_2.html#K3">cod</a> <font color="Maroon">f'</font>

;<br/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>
 ;<br/><b>then </b> <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">g'</font> = 
 <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">f'</font>
<b>by </b><i/>
<br/>.= 
 <a href="mod_2.html#K3">cod</a> <font color="Maroon">f'</font>

;<br/><b>hence </b><a NAME="E3:31_3"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="modcat_1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_4"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:31_4"/><b>then </b>
 <a href="modcat_1.html#K5" target="_self">dom'</a> <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a>  <a href="modcat_1.html#K6" target="_self">cod'</a> <font color="Maroon">f'</font>
 <b>by </b><i>, <a class="ref" href="modcat_1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:31_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:31"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">UN</font>,<font color="Maroon">R</font></span>)</span> iff  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/>


</div>
