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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> ; <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> being <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="b1">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> holds <br/><font color="Olive" title="b2">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/><b>let </b><font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a> <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="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies <font color="Maroon" title="c3">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:131"/><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="E2:131"/><i><font color="Green" title="E109">A2</font></i>: 
<font color="Maroon" title="c2">G</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a></span><br/>

<b>consider </b><font color="Maroon" title="c4">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="E4:131"/><i><font color="Green" title="E110">A3</font></i>: 
<font color="Maroon" title="c4">g</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="#E1:131"><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/>
<a NAME="E5:131"/><i><font color="Green" title="E111">A4</font></i>: 
( <font color="Maroon" title="c4">g</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> &amp; <font color="Maroon" title="c4">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="#E4:131"><i><font color="Green" title="E110">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:131"/><i><font color="Green" title="E112">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:131"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">GROUP_6:def 2</a></i>;<br/>
<a NAME="E8:131"/>
<font color="Maroon" title="c5">e</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 <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="#E7:131"><i><font color="Green" title="E112">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:131"/><b>then </b><i><font color="Green" title="E113">A6</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">H</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K14" title="GROUP_9:func.14">Image</a> <font color="Maroon" title="c4">g</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:131"><i><font color="Green" title="E111">A4</font></i></a>, <a class="ref" href="group_9.html#T51" target="_self" title="GROUP_9:th.51">Th51</a></i>
<br/>.= 
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="#E7:131"><i><font color="Green" title="E112">A5</font></i></a>, <a class="ref" href="group_9.html#D25" target="_self" title="GROUP_9:def.25">Def25</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E9:131"><i><font color="Green" title="E113">A6</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
;<br/>
<b>hence </b><a NAME="E11:131"/>
<font color="Maroon" title="c3">H</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 <b>by </b><i><a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">GROUP_6:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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