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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AddGroup</a>;<br/>

<b>set </b><font color="Maroon" title="c2">i</font> =  <a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>thus </b><a NAME="E1:50"/>
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>thus </b><a NAME="E2:50"/>
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>thus </b><a NAME="E3:50"/>
for <font color="Olive" title="b1">f</font> being <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>  st  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</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="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f</font> be  <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/>

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">H</font> =  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c3">f</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="grcat_1.html#E1:50_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="grcat_1.html#D19" target="_self" title="GRCAT_1:def.19">Def19</a></i>;<br/>
<a NAME="E3:50_1"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c6">m</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:50_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <span class="p1">(<span class="default"><a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">m</font> #)
 <b>by </b><i><a class="ref" href="grcat_1.html#T23" target="_self" title="GRCAT_1:th.23">Th23</a></i>;<br/>
<a NAME="E6:50_1"/>
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E7:50_1"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:50_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="grcat_1.html#E3:50_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="grcat_1.html#E5:50_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="grcat_1.html#D21" target="_self" title="GRCAT_1:def.21">Def21</a></i>;<br/>


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

<b>thus </b><a NAME="E4:50"/>
for <font color="Olive" title="b1">g</font> being <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>  st  <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">g</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</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="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">f</font> be  <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/>

<b>assume </b><a NAME="E1:50_2"/><i><font color="Green" title="E29">A4</font></i>: 
 <a href="grcat_1.html#K8" title="GRCAT_1:func.8">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">H</font> =  <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c3">f</font> as    <a href="grcat_1.html#M1" title="GRCAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c1">G</font>, <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font> <b>by </b><i><a class="txt" href="grcat_1.html#E1:50_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="grcat_1.html#D19" target="_self" title="GRCAT_1:def.19">Def19</a></i>;<br/>
<a NAME="E3:50_2"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c6">m</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font></span>)</span><b> such that </b><br/><a NAME="E5:50_2"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c5">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="grcat_1.html#G1" title="GRCAT_1:aggr.1">GroupMorphismStr</a>(# <font color="Maroon" title="c1">G</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K9" title="GRCAT_1:func.9">cod</a> <font color="Maroon" title="c3">f</font></span>)</span>,<font color="Maroon" title="c6">m</font> #)
 <b>by </b><i><a class="ref" href="grcat_1.html#T23" target="_self" title="GRCAT_1:th.23">Th23</a></i>;<br/>
<a NAME="E6:50_2"/>
<font color="Maroon" title="c6">m</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">m</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E7:50_2"/>
<font color="Maroon" title="c3">f</font> <a href="grcat_1.html#K14" title="GRCAT_1:func.14">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K12" title="GRCAT_1:func.12">ID</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:50_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="grcat_1.html#E3:50_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="grcat_1.html#E5:50_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="grcat_1.html#D21" target="_self" title="GRCAT_1:def.21">Def21</a></i>;<br/>


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


</div>
