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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="mod_1.html#NM1">AddGroup</a>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="mod_1.html#NM1">AddGroup</a>;<br/>





<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">D</font> is    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> implies for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Olive">x</font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:67"/>
( ( for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Olive">x</font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> ) implies <font color="Maroon">D</font> is    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> holds  <font color="Olive">x</font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:67_1"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">x</font> is  <a href="grcat_1.html#V2" target="_self">strict</a> <a href="grcat_1.html#NM1">GroupMorphism</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">D'</font> = <font color="Maroon">D</font> as   <a href="grcat_1.html#NM5">GroupMorphism_DOMAIN</a> <b>by </b><i/>;<br/>
<a NAME="E4:67_1"/>
for <font color="Olive">x</font> being   <a href="grcat_1.html#M3" target="_self">Element</a> of <font color="Maroon">D'</font> holds  <font color="Olive">x</font> is  <a href="grcat_1.html#V2" target="_self">strict</a>  <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:67_1"/>
<font color="Maroon">D</font> is    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


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


</div>
