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<b>let </b><font color="Maroon" title="c1">H</font>, <font color="Maroon" title="c2">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">g</font> being  <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">H</font>  st <font color="Maroon" title="c2">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> holds <br/> <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Olive" title="b1">g</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a></span><br/><b>let </b><font color="Maroon" title="c3">g</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c1">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="c2">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> implies  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> )</span><br/>





<b>assume </b><a NAME="E1:70"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c2">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</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">  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a></span><br/>

<b>let </b><font color="Maroon" title="c4">h1</font>, <font color="Maroon" title="c5">h2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h1</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c4">h1</font>, <font color="Maroon" title="c7">d</font> = <font color="Maroon" title="c5">h2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">H</font> <b>by </b><i><a class="ref" href="group_2.html#T51" title="GROUP_2:th.51">GROUP_2:51</a></i>;<br/>
<a NAME="E3:70"/>
<font color="Maroon" title="c4">h1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E5:70"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c4">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></i>;<br/>
<a NAME="E6:70"/>
<font color="Maroon" title="c5">h2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c3">g</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E8:70"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c5">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="ref" href="group_6.html#T54" target="_self" title="GROUP_6:th.54">Th54</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font> = 
<font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">d</font>
<b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:70"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E8:70"><i><font color="Green" title="E46">A3</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">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:70"><i><font color="Green" title="E44">A1</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="c7">d</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="txt" href="#E5:70"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="#E8:70"><i><font color="Green" title="E46">A3</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="c5">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h1</font>
<b>by </b><i><a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</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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