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

<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">H</font>, <font color="Olive" title="b3">I</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>  st <font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">H</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  &amp; <font color="Olive" title="b2">H</font>,<font color="Olive" title="b3">I</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a>  holds <br/><font color="Olive" title="b1">G</font>,<font color="Olive" title="b3">I</font> <a href="group_9.html#R3" title="GROUP_9:pred.3">are_isomorphic</a> </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  &amp; <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>  implies <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>  )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:128"/><i><font color="Green" title="E105">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="E2:128"/><i><font color="Green" title="E106">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> </span><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="E4:128"/><i><font color="Green" title="E107">A3</font></i>: 
<font color="Maroon" title="c5">g</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="#E1:128"><i><font color="Green" title="E105">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="E6:128"/><i><font color="Green" title="E108">A4</font></i>: 
<font color="Maroon" title="c6">h1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="#E2:128"><i><font color="Green" title="E106">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="E7:128"/>
( <font color="Maroon" title="c5">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &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="txt" href="#E4:128"><i><font color="Green" title="E107">A3</font></i></a></i>;<br/>
<a NAME="E8:128"/><b>then </b><i><font color="Green" title="E109">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="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E9:128"/>
( <font color="Maroon" title="c6">h1</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &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="txt" href="#E6:128"><i><font color="Green" title="E108">A4</font></i></a></i>;<br/>
<a NAME="E10:128"/><b>then </b><i><font color="Green" title="E110">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="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E11:128"/>
 <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="#E8:128"><i><font color="Green" title="E109">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="E12:128"/><b>then </b><i><font color="Green" title="E111">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="#E10:128"><i><font color="Green" title="E110">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="E13:128"/>
<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="#E8:128"><i><font color="Green" title="E109">A5</font></i></a>, <a class="txt" href="#E10:128"><i><font color="Green" title="E110">A6</font></i></a></i>;<br/>
<a NAME="E14:128"/><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> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</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> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="#E12:128"><i><font color="Green" title="E111">A7</font></i></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E15:128"/><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> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
;<br/>
<b>hence </b><a NAME="E16:128"/>
<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
