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<b>let </b><font color="Maroon">H</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><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/>





<a NAME="E1:65"/>
<font color="Maroon">h</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<a NAME="E1:65_1"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:65_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:65_1"/>
<font color="Maroon">h</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>now </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/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:65"/>
<font color="Maroon">h</font> is   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" target="_self">Image</a> <font color="Maroon">h</font></span>)</span>
 <b>by </b><i/>;<br/>


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