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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:35"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<a NAME="E2:35"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green">E29</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E4:35"/><i><font color="Green">E30</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:35"/><i><font color="Green">E31</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>thus </b><a NAME="E6:35"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> iff  <a href="mod_2.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modcat_1.html#E3:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>

<b>thus </b><a NAME="E7:35"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="modcat_1.html#K11" target="_self">comp</a> <span class="p2">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E2:35"><i><font color="Green">E28</font></i></a>, <a class="txt" href="modcat_1.html#E3:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="modcat_1.html#T13" target="_self">Th13</a></i>;<br/>

<b>thus </b><a NAME="E8:35"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="mod_2.html#K8">*</a> <font color="Maroon">c<sub>8</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E33</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="modcat_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><a href="modcat_1.html#K11" target="_self">comp</a> <span class="p2">(<span class="default"><a href="modcat_1.html#K3" target="_self">LModObjects</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="mod_2.html#K8">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="modcat_1.html#E7:35"><i><font color="Green">E32</font></i></a>, <a class="txt" href="modcat_1.html#E1:35_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="modcat_1.html#D14" target="_self">Def14</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E9:35"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> iff  <a href="mod_2.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modcat_1.html#E4:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>

<b>thus </b><a NAME="E10:35"/>
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font> iff  <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="mod_2.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="modcat_1.html#E1:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="modcat_1.html#E5:35"><i><font color="Green">E31</font></i></a>, <a class="ref" href="modcat_1.html#T17" target="_self">Th17</a></i>;<br/>


</div>
