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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c1">G</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E2:6_1"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
 <b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/><a NAME="E3:6_1"/>
<font color="Maroon" title="c2">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="autgroup.html#E1:6_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/><a NAME="E4:6_1"/><b>then </b>
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D12" title="GROUP_6:def.12">GROUP_6:def 12</a></i>;<br/><b>hence </b><a NAME="E5:6_1"/>
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="autgroup.html#E2:6_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:6_2"/>
<font color="Maroon" title="c2">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 ;<br/><a NAME="E2:6_2"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c2">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a>  &amp; <font color="Maroon" title="c2">h</font> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a>  )
 <b>by </b><i><a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/><a NAME="E3:6_2"/><b>then </b>
<font color="Maroon" title="c2">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="group_6.html#D12" title="GROUP_6:def.12">GROUP_6:def 12</a></i>;<br/><b>hence </b><a NAME="E4:6_2"/>
<font color="Maroon" title="c2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="autgroup.html#K1" title="AUTGROUP:func.1">Aut</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="autgroup.html#E2:6_2"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="autgroup.html#D1" target="_self" title="AUTGROUP:def.1">Def1</a></i>;<br/></div>
<b>end;</b></div>


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