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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>] <b>means </b> <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">g</font> be   <a href="grcat_1.html#NM1">GroupMorphism</a>, <font color="Maroon">f</font> be   <a href="grcat_1.html#NM1">GroupMorphism</a>;<br/>

<b>assume </b><a NAME="E1:45"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">f</font>]
 ;<br/>

<b>consider </b><font color="Maroon">G2</font> being   <a href="mod_1.html#NM1">AddGroup</a>, <font color="Maroon">G3</font> being   <a href="mod_1.html#NM1">AddGroup</a><b> such that </b><br/><a NAME="E3:45"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">g</font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:45"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">G2</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K9" target="_self">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">G1</font> being   <a href="mod_1.html#NM1">AddGroup</a>, <font color="Maroon">G2'</font> being   <a href="mod_1.html#NM1">AddGroup</a><b> such that </b><br/><a NAME="E6:45"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2'</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:45"/>
<font color="Maroon">G2'</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K10" target="_self">cod</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:45"/>
 ex <font color="Olive">G1</font>, <font color="Olive">G2</font>, <font color="Olive">G3</font> being  <a href="mod_1.html#NM1">AddGroup</a> st <br/>( <font color="Maroon">g</font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Olive">G2</font>,<font color="Olive">G3</font> &amp; <font color="Maroon">f</font> is    <a href="grcat_1.html#M1" target="_self">Morphism</a> of <font color="Olive">G1</font>,<font color="Olive">G2</font> )
 <b>by </b><i>, , , </i>;<br/>


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