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<b>let </b><font color="Maroon" title="c1">H</font>, <font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">H</font>;<br/>





<a NAME="E1:66"/>
<font color="Maroon" title="c3">h</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<a NAME="E1:66_1"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T53" target="_self" title="GROUP_6:th.53">Th53</a></i>;<br/>
<a NAME="E2:66_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:66_1"/>
<font color="Maroon" title="c3">h</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_6.html#E1:66_1"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">f'</font> = <font color="Maroon" title="c3">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon" title="c4">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:66"/>
<font color="Maroon" title="c3">h</font> is   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>;<br/>


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