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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">H</font> being <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>  st <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">H</font> <a href="group_6.html#R1" title="GROUP_6:pred.1">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> holds <br/><font color="Olive" title="b1">H</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a></span><br/><b>let </b><font color="Maroon" title="c2">H</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R1" title="GROUP_6:pred.1">are_isomorphic</a>  &amp; <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> implies <font color="Maroon" title="c2">H</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:101"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font> <a href="group_6.html#R1" title="GROUP_6:pred.1">are_isomorphic</a> 
 <b>and </b><br/><a NAME="E2:101"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</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">H</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a></span><br/>

<b>consider </b><font color="Maroon" title="c3">h</font> being   <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="c2">H</font><b> such that </b><br/><a NAME="E4:101"/><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="group_6.html#E1:101"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="group_6.html#D15" target="_self" title="GROUP_6:def.15">Def15</a></i>;<br/>
<a NAME="E5:101"/><i><font color="Green" title="E67">A4</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="group_6.html#NR2" title="GROUP_6:NR.2">is_epimorphism</a> 
 
<b>by </b><i><a class="txt" href="group_6.html#E4:101"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="group_6.html#D14" target="_self" title="GROUP_6:def.14">Def14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">c</font></span><br/><b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E2:101_1"/><i><font color="Green" title="E68">A5</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="group_6.html#E5:101"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="group_6.html#T68" target="_self" title="GROUP_6:th.68">Th68</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:101_1"/><i><font color="Green" title="E69">A6</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <b>by </b><i><a class="txt" href="group_6.html#E5:101"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="group_6.html#T68" target="_self" title="GROUP_6:th.68">Th68</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">d</font> = 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E2:101_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="group_6.html#E4:101_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span>)</span>
<b>by </b><i><a class="txt" href="group_6.html#E2:101"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="group_6.html#E72"><i><font color="Green" title="E43">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="group_6.html#E2:101_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="group_6.html#E4:101_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="group_6.html#D7" target="_self" title="GROUP_6:def.7">Def7</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><b>end;</b></div>
<b>hence </b><a NAME="E7:101"/>
<font color="Maroon" title="c2">H</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>
 <b>by </b><i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</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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