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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="grcat_1.html#NM1">GroupMorphism</a>,  <a href="grcat_1.html#NM1">GroupMorphism</a>] means  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="grcat_1.html#NM1">GroupMorphism</a>;<br/>

<b>assume </b><a NAME="E1:45"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="mod_1.html#NM1">AddGroup</a><b> such that </b><br/><a NAME="E3:45"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E3:45"><i><font color="Green">E26</font></i></a>, <a class="ref" href="grcat_1.html#D19" target="_self">Def19</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> being   <a href="mod_1.html#NM1">AddGroup</a><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="grcat_1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E7:45"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="grcat_1.html#E6:45"><i><font color="Green">E28</font></i></a>, <a class="ref" href="grcat_1.html#D19" target="_self">Def19</a></i>;<br/>
<b>hence </b><a NAME="E8:45"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="mod_1.html#NM1">AddGroup</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:45"><i><font color="Green">E25</font></i></a>, <a class="txt" href="grcat_1.html#E3:45"><i><font color="Green">E26</font></i></a>, <a class="txt" href="grcat_1.html#E4:45"><i><font color="Green">E27</font></i></a>, <a class="txt" href="grcat_1.html#E6:45"><i><font color="Green">E28</font></i></a></i>;<br/>


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