<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/><b>let </b><font color="Maroon" title="c4">h</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:72"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c4">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a> 
 <b>and </b><br/><a NAME="E2:72"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c3">c</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a>  <a href="group_6.html#K12" title="GROUP_6:func.12">Image</a> <font color="Maroon" title="c4">h</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">h'</font> = <font color="Maroon" title="c4">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">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="c4">h</font></span>)</span> <b>by </b><i><a class="ref" href="group_6.html#T58" target="_self" title="GROUP_6:th.58">Th58</a></i>;<br/>
<a NAME="E4:72"/><i><font color="Green" title="E55">A3</font></i>: 
<font color="Maroon" title="c5">h'</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="group_6.html#E1:72"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="group_6.html#D12" target="_self" title="GROUP_6:def.12">Def12</a></i>;<br/>
<a NAME="E5:72"/>
<font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>proof </b></a><div class="add">

<a NAME="E1:72_1"/><i><font color="Green" title="E56">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">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="c4">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="E2:72_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_6.html#E2:72"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="rlvect_1.html#D1" title="RLVECT_1:def.1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:72_1"/>
<font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">h'</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="group_6.html#E4:72"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="group_6.html#E1:72_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:72"/>
<font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 ;<br/>


</div>
