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

<b>let </b><font color="Maroon">G</font> be   <a href="mod_1.html#NM1">AddGroup</a>;<br/>

<b>set </b><font color="Maroon">i</font> =  <a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font>;<br/>
<b>thus </b><a NAME="E1:54"/>
 <a href="grcat_1.html#K9" target="_self">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 ;<br/>

<b>thus </b><a NAME="E2:54"/>
 <a href="grcat_1.html#K10" target="_self">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 ;<br/>

<b>thus </b><a NAME="E3:54"/>
for <font color="Olive">f</font> being <a href="grcat_1.html#V2" target="_self">strict</a> <a href="grcat_1.html#NM1">GroupMorphism</a>  st  <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="grcat_1.html#V2" target="_self">strict</a> <a href="grcat_1.html#NM1">GroupMorphism</a>;<br/>

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E29</font></i>: 
 <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">H</font> =  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">f</font>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="grcat_1.html#M1" target="_self">Morphism</a> of  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">f</font>,<font color="Maroon">G</font> <b>by </b><i>, </i>;<br/>
<a NAME="E3:54_1"/><i><font color="Green">E42</font></i>: 
 <a href="grcat_1.html#K9" target="_self">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 
;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:54_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <span class="p1">(<span class="default"><a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">G</font>,<font color="Maroon">m</font> #)
 <b>by </b><i/>;<br/>
<a NAME="E6:54_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7" target="_self">id</a> <font color="Maroon">G</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">m</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:54_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="grcat_1.html#K15" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="grcat_1.html#T4" target="_self">Th4</a>, </i>;<br/>


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

<b>thus </b><a NAME="E4:54"/>
for <font color="Olive">g</font> being <a href="grcat_1.html#V2" target="_self">strict</a> <a href="grcat_1.html#NM1">GroupMorphism</a>  st  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> holds <br/><font color="Olive">g</font> <a href="grcat_1.html#K15" target="_self">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be  <a href="grcat_1.html#V2" target="_self">strict</a> <a href="grcat_1.html#NM1">GroupMorphism</a>;<br/>

<b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E44</font></i>: 
 <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 ;<br/>

<b>set </b><font color="Maroon">H</font> =  <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">f</font>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>, <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">f</font> <b>by </b><i>, </i>;<br/>
<a NAME="E3:54_2"/><i><font color="Green">E90</font></i>: 
 <a href="grcat_1.html#K10" target="_self">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 
;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">f</font></span>)</span><b> such that </b><br/><a NAME="E5:54_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#G1" target="_self">GroupMorphismStr</a>(# <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">m</font> #)
 <b>by </b><i/>;<br/>
<a NAME="E6:54_2"/>
<font color="Maroon">m</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" target="_self">id</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:54_2"/>
<font color="Maroon">f</font> <a href="grcat_1.html#K15" target="_self">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K13" target="_self">ID</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="grcat_1.html#T4" target="_self">Th4</a>, </i>;<br/>


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


</div>
