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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="GROUP_9:NM.1">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/><span class="kw">let </span><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#NM1" title="GROUP_9:NM.1">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:127"/><span class="lab"><font color="Green" title="E105">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:127"/><span class="lab"><font color="Green" title="E106">A2</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><font color="Maroon" title="c5">g</font> being   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c3">H</font><span class="kw"> such that </span><br/><a NAME="E4:127"/><span class="lab"><font color="Green" title="E107">A3</font></span>: 
<font color="Maroon" title="c5">g</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:127"><span class="lab"><font color="Green" title="E105">A1</font></span></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">h1</font> being   <a href="group_9.html#NM3" title="GROUP_9:NM.3">Homomorphism</a> of <font color="Maroon" title="c3">H</font>,<font color="Maroon" title="c4">I</font><span class="kw"> such that </span><br/><a NAME="E6:127"/><span class="lab"><font color="Green" title="E108">A4</font></span>: 
<font color="Maroon" title="c6">h1</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:127"><span class="lab"><font color="Green" title="E106">A2</font></span></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def22</a></span>;<br/>
<a NAME="E7:127"/><span class="lab"><font color="Green" title="E109">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E108">A4</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E8:127"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E107">A3</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E9:127"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:127"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:127"><span class="lab"><font color="Green" title="E109">A5</font></span></a>, <a class="ref" href="relat_1.html#T28" title="RELAT_1:th.28">RELAT_1:28</a></span>;<br/>
<a NAME="E11:127"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:127"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E107">A3</font></span></a>, <a class="txt" href="#E6:127"><span class="lab"><font color="Green" title="E108">A4</font></span></a>, <a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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