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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="mod_1.html#NM1">AddGroup</a>;<br/>





<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">c<sub>1</sub></font> is    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="grcat_1.html#D24" target="_self">Def24</a></i>;<br/>

<b>thus </b><a NAME="E2:67"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> ) implies <font color="Maroon">c<sub>1</sub></font> is    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:67_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="grcat_1.html#NM5">GroupMorphism_DOMAIN</a> <b>by </b><i><a class="ref" href="grcat_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E4:67_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="grcat_1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>4</sub></font> holds <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E1:67_1"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:67_1"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="grcat_1.html#M4" target="_self">GroupMorphism_DOMAIN</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="grcat_1.html#D24" target="_self">Def24</a></i>;<br/>


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


</div>
