<?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">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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>reconsider </b><font color="Maroon" title="c6">b'</font> = <font color="Maroon" title="c5">b</font> as    <a href="grcat_1.html#M2" title="GRCAT_1:mode.2">Element</a> of  <a href="grcat_1.html#K19" title="GRCAT_1:func.19">GroupObjects</a> <font color="Maroon" title="c1">UN</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b''</font> = <font color="Maroon" title="c6">b'</font> as   <a href="mod_1.html#NM1" title="MOD_1:NM.1">AddGroup</a> ;<br/>
<a NAME="E3:99"/><i><font color="Green" title="E61">A1</font></i>: 
 <a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#K23" title="GRCAT_1:func.23">ID</a> <font color="Maroon" title="c6">b'</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T48" target="_self" title="GRCAT_1:th.48">Th48</a></i>;<br/>
<b>hence </b><i><font color="Green" title="E62">A2</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" title="GRCAT_1:func.13">ID</a> <font color="Maroon" title="c7">b''</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font>

;<br/><br/>

<b>thus </b><i><font color="Green" title="E63">A3</font></i>:  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
 <a href="grcat_1.html#K10" title="GRCAT_1:func.10">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" title="GRCAT_1:func.13">ID</a> <font color="Maroon" title="c7">b''</font></span>)</span>
<b>by </b><i><a class="txt" href="grcat_1.html#E3:99"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="grcat_1.html#T49" target="_self" title="GRCAT_1:th.49">Th49</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font>

;<br/>

<b>thus </b><a NAME="E6:99"/>
for <font color="Olive" title="b1">f</font> being  <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>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</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>assume </b><a NAME="E1:99_3"/><i><font color="Green" title="E64">A4</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c8">f</font> as  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <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> <b>by </b><i><a class="ref" href="grcat_1.html#T47" target="_self" title="GRCAT_1:th.47">Th47</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f'</font> = <font color="Maroon" title="c9">f1</font> as  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a> <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a> ;<br/>
<a NAME="E4:99_3"/><i><font color="Green" title="E65">A5</font></i>: 
 <a href="grcat_1.html#K10" title="GRCAT_1:func.10">cod</a> <font color="Maroon" title="c10">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b''</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:99_3"><i><font color="Green" title="E64">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><span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c8">f</font> = 
<span class="p1">(<span class="default"><a href="grcat_1.html#K13" title="GRCAT_1:func.13">ID</a> <font color="Maroon" title="c7">b''</font></span>)</span> <a href="grcat_1.html#K15" title="GRCAT_1:func.15">*</a> <font color="Maroon" title="c10">f'</font>
<b>by </b><i><a class="txt" href="grcat_1.html#E3:99"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="grcat_1.html#E4:99"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="grcat_1.html#E1:99_3"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="grcat_1.html#T50" target="_self" title="GRCAT_1:th.50">Th50</a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font>
<b>by </b><i><a class="txt" href="grcat_1.html#E4:99_3"><i><font color="Green" title="E65">A5</font></i></a>, <a class="ref" href="grcat_1.html#T34" target="_self" title="GRCAT_1:th.34">Th34</a></i>
;<br/>


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

<b>thus </b><a NAME="E7:99"/>
for <font color="Olive" title="b1">g</font> being  <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>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> holds <br/><font color="Olive" title="b1">g</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</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>assume </b><a NAME="E1:99_4"/><i><font color="Green" title="E64">A6</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c8">f</font> as  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a>  <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> <b>by </b><i><a class="ref" href="grcat_1.html#T47" target="_self" title="GRCAT_1:th.47">Th47</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f'</font> = <font color="Maroon" title="c9">f1</font> as  <a href="grcat_1.html#V2" title="GRCAT_1:attr.2">strict</a> <a href="grcat_1.html#NM1" title="GRCAT_1:NM.1">GroupMorphism</a> ;<br/>
<a NAME="E4:99_4"/><i><font color="Green" title="E65">A7</font></i>: 
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">dom</a> <font color="Maroon" title="c10">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b''</font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:99_4"><i><font color="Green" title="E64">A6</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><font color="Maroon" title="c8">f</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5" title="CAT_1:func.5">id</a> <font color="Maroon" title="c5">b</font></span>)</span> = 
<font color="Maroon" title="c10">f'</font> <a href="grcat_1.html#K15" title="GRCAT_1:func.15">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" title="GRCAT_1:func.13">ID</a> <font color="Maroon" title="c7">b''</font></span>)</span>
<b>by </b><i><a class="txt" href="grcat_1.html#E3:99"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="grcat_1.html#E5:99"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="grcat_1.html#E1:99_4"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="grcat_1.html#T50" target="_self" title="GRCAT_1:th.50">Th50</a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font>
<b>by </b><i><a class="txt" href="grcat_1.html#E4:99_4"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="grcat_1.html#T34" target="_self" title="GRCAT_1:th.34">Th34</a></i>
;<br/>


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


</div>
