<?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>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="grcat_1.html#K28" target="_self">GroupCat</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="grcat_1.html#K19" target="_self">GroupObjects</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="grcat_1.html#K20" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K19" target="_self">GroupObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K28" target="_self">GroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be    <a href="grcat_1.html#M3" target="_self">Element</a> of  <a href="grcat_1.html#K20" target="_self">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K19" target="_self">GroupObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:98"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:98"/><b>then </b><i><font color="Green">E59</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="grcat_1.html#K21" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E3:98"/><i><font color="Green">E60</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K22" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E4:98"/><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K21" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>
<a NAME="E5:98"/><i><font color="Green">E62</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="grcat_1.html#K22" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>
<b>thus </b><a NAME="E6:98"/>
(  <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#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> iff  <a href="grcat_1.html#K21" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K22" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="txt" href="grcat_1.html#E3:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>

<b>thus </b><a NAME="E7:98"/><i><font color="Green">E63</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="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> iff <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</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="grcat_1.html#K27" target="_self">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K19" target="_self">GroupObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E2:98"><i><font color="Green">E59</font></i></a>, <a class="txt" href="grcat_1.html#E3:98"><i><font color="Green">E60</font></i></a>, <a class="ref" href="grcat_1.html#D35" target="_self">Def35</a></i>;<br/>

<b>thus </b><a NAME="E8:98"/>
(  <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#K3">cod</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="98_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:98_1"/><i><font color="Green">E64</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="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:98_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</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="grcat_1.html#K28" target="_self">GroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T46" target="_self">Th46</a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>6</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="grcat_1.html#K27" target="_self">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K19" target="_self">GroupObjects</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="grcat_1.html#E7:98"><i><font color="Green">E63</font></i></a>, <a class="txt" href="grcat_1.html#E1:98_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="grcat_1.html#D35" target="_self">Def35</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E9:98"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font> iff  <a href="grcat_1.html#K21" target="_self">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K21" target="_self">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="txt" href="grcat_1.html#E4:98"><i><font color="Green">E61</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>

<b>thus </b><a NAME="E10:98"/>
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</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="grcat_1.html#K22" target="_self">cod</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K22" target="_self">cod</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:98"><i><font color="Green">E58</font></i></a>, <a class="txt" href="grcat_1.html#E5:98"><i><font color="Green">E62</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self">Th49</a></i>;<br/>


</div>
