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<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font>, <font color="Maroon" title="c4">I</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:132"/><i><font color="Green" title="E108">A1</font></i>: 
<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:132"/><i><font color="Green" title="E109">A2</font></i>: 
<font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">I</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">g</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font><b> such that </b><br/><a NAME="E4:132"/><i><font color="Green" title="E110">A3</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="group_9.html#NR4" title="GROUP_9:NR.4">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E1:132"><i><font color="Green" title="E108">A1</font></i></a>, <a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">Def22</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">h1</font> being   <a href="group_9.html#NM4" title="GROUP_9:NM.4">Homomorphism</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">I</font><b> such that </b><br/><a NAME="E6:132"/><i><font color="Green" title="E111">A4</font></i>: 
<font color="Maroon" title="c6">h1</font> <a href="group_9.html#NR4" title="GROUP_9:NR.4">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E2:132"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">Def22</a></i>;<br/>
<a NAME="E7:132"/>
( <font color="Maroon" title="c5">g</font> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="group_9.html#NR2" title="GROUP_9:NR.2">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="group_9.html#E4:132"><i><font color="Green" title="E110">A3</font></i></a>, <a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def21</a></i>;<br/>
<a NAME="E8:132"/><b>then </b><i><font color="Green" title="E112">A5</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</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="c3">H</font> &amp; <font color="Maroon" title="c5">g</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_9.html#D19" target="_self" title="GROUP_9:def.19">Def19</a>, <a class="ref" href="group_9.html#D20" target="_self" title="GROUP_9:def.20">Def20</a></i>;<br/>
<a NAME="E9:132"/>
( <font color="Maroon" title="c6">h1</font> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  &amp; <font color="Maroon" title="c6">h1</font> <a href="group_9.html#NR2" title="GROUP_9:NR.2">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="group_9.html#E6:132"><i><font color="Green" title="E111">A4</font></i></a>, <a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def21</a></i>;<br/>
<a NAME="E10:132"/><b>then </b><i><font color="Green" title="E113">A6</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h1</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="c4">I</font> &amp; <font color="Maroon" title="c6">h1</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_9.html#D19" target="_self" title="GROUP_9:def.19">Def19</a>, <a class="ref" href="group_9.html#D20" target="_self" title="GROUP_9:def.20">Def20</a></i>;<br/>
<a NAME="E11:132"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E8:132"><i><font color="Green" title="E112">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:132"/><b>then </b><i><font color="Green" title="E114">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c5">g</font></span>)</span> <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="c4">I</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E10:132"><i><font color="Green" title="E113">A6</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/>
<a NAME="E13:132"/>
<font color="Maroon" title="c6">h1</font> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c5">g</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_9.html#E8:132"><i><font color="Green" title="E112">A5</font></i></a>, <a class="txt" href="group_9.html#E10:132"><i><font color="Green" title="E113">A6</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/>
<a NAME="E14:132"/><b>then </b>
( <font color="Maroon" title="c6">h1</font> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c5">g</font> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  &amp; <font color="Maroon" title="c6">h1</font> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c5">g</font> <a href="group_9.html#NR2" title="GROUP_9:NR.2">is_monomorphism</a>  )
 
<b>by </b><i><a class="txt" href="group_9.html#E12:132"><i><font color="Green" title="E114">A7</font></i></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def19</a>, <a class="ref" href="group_9.html#D20" target="_self" title="GROUP_9:def.20">Def20</a></i>;<br/>
<a NAME="E15:132"/><b>then </b>
<font color="Maroon" title="c6">h1</font> <a href="group_9.html#K11" title="GROUP_9:func.11">*</a> <font color="Maroon" title="c5">g</font> <a href="group_9.html#NR4" title="GROUP_9:NR.4">is_isomorphism</a> 
 
<b>by </b><i><a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def21</a></i>;<br/>
<b>hence </b><a NAME="E16:132"/>
<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">I</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="group_9.html#D22" target="_self" title="GROUP_9:def.22">Def22</a></i>;<br/>


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