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<b>let </b><font color="Maroon">g</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>, <font color="Maroon">h</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">H</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:51_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E2:51_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E3:51_1_2"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<a NAME="E1:51_1_2_1"/>
( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:51_1_2_1"/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:51_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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