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

<b>let </b><font color="Maroon" title="c1">UN</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/>

<b>set </b><font color="Maroon" title="c2">C</font> =  <a href="grcat_1.html#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c3">V</font> =  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">GroupObjects</a> <font color="Maroon" title="c1">UN</font>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">f'</font>, <font color="Maroon" title="c8">g'</font> be    <a href="grcat_1.html#M3" title="GRCAT_1:mode.3">Element</a> of  <a href="grcat_1.html#K20" title="GRCAT_1:func.20">Morphs</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E58">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f'</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> )
 ;<br/>

<a NAME="E2:96"/><b>then </b><i><font color="Green" title="E59">A2</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">dom</a> <font color="Maroon" title="c8">g'</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>
<a NAME="E3:96"/><i><font color="Green" title="E60">A3</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K22" title="GRCAT_1:func.22">cod</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>
<a NAME="E4:96"/><i><font color="Green" title="E61">A4</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">dom</a> <font color="Maroon" title="c7">f'</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>
<a NAME="E5:96"/><i><font color="Green" title="E62">A5</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K22" title="GRCAT_1:func.22">cod</a> <font color="Maroon" title="c8">g'</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>
<b>thus </b><a NAME="E6:96"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> iff  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">dom</a> <font color="Maroon" title="c8">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K22" title="GRCAT_1:func.22">cod</a> <font color="Maroon" title="c7">f'</font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="grcat_1.html#E3:96"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>

<b>thus </b><a NAME="E7:96"/><i><font color="Green" title="E63">A6</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K27" title="GRCAT_1:func.27">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E2:96"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="grcat_1.html#E3:96"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="grcat_1.html#D35" target="_self" title="GRCAT_1:def.35">Def35</a></i>;<br/>

<b>thus </b><a NAME="E8:96"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K15" title="GRCAT_1:func.15">*</a> <font color="Maroon" title="c7">f'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E64">A7</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font>
 ;<br/>

<a NAME="E2:96_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c5">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T46" target="_self" title="GRCAT_1:th.46">Th46</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><a href="grcat_1.html#K27" title="GRCAT_1:func.27">comp</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K19" title="GRCAT_1:func.19">GroupObjects</a> <font color="Maroon" title="c1">UN</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g'</font>,<font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="cat_1.html#D4" title="CAT_1:def.4">CAT_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">g'</font> <a href="grcat_1.html#K15" title="GRCAT_1:func.15">*</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="grcat_1.html#E7:96"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="grcat_1.html#E1:96_1"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="grcat_1.html#D35" target="_self" title="GRCAT_1:def.35">Def35</a></i>
;<br/><br/>


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

<b>thus </b><a NAME="E9:96"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c6">g</font> iff  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">dom</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K21" title="GRCAT_1:func.21">dom</a> <font color="Maroon" title="c8">g'</font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="grcat_1.html#E4:96"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>

<b>thus </b><a NAME="E10:96"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c6">g</font> iff  <a href="grcat_1.html#K22" title="GRCAT_1:func.22">cod</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K22" title="GRCAT_1:func.22">cod</a> <font color="Maroon" title="c8">g'</font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:96"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="grcat_1.html#E5:96"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>;<br/>


</div>
