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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ;<br/><b>consider </b><font color="Maroon" title="c4">G'</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><b> such that </b><br/><a NAME="E3:14_1_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G'</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">G'</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">G</font> #)
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">a</font> = the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> as   <a href="group_9.html#NM1" title="GROUP_9:NM.1">Action</a> of <font color="Maroon" title="c1">O</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E3:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><a NAME="E5:14_1_1_1"/>
<font color="Maroon" title="c5">a</font> is <a href="group_9.html#V1" title="GROUP_9:attr.1">distributive</a>
 <b>by </b><i><a class="txt" href="group_9.html#E3:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="group_9.html#D5" target="_self" title="GROUP_9:def.5">Def5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">f'</font> = <font color="Maroon" title="c5">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c4">G'</font>,<font color="Maroon" title="c4">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E1:14_1_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="group_9.html#D4" target="_self" title="GROUP_9:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f'</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E3:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">g1</font>, <font color="Maroon" title="c9">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">g1'</font> = <font color="Maroon" title="c8">g1</font>, <font color="Maroon" title="c11">g2'</font> = <font color="Maroon" title="c9">g2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G'</font> <b>by </b><i><a class="txt" href="group_9.html#E3:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">g2</font></span>)</span> = 
<font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">g1'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">g2'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E3:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g1'</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g2'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">G</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E3:14_1_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:14_1_1_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g2</font></span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:14_1_1_1"/>
the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font> is   <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="c2">G</font>
 <b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:14_1_1"/>
( ( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> implies the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">o</font> is   <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="c2">G</font> ) &amp; ( not <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> implies  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> is   <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="c2">G</font> ) &amp; ( for <font color="Olive">b<sub>1</sub></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="c2">G</font> holds  verum ) )
 <b>by </b><i><a class="ref" href="group_6.html#T47" title="GROUP_6:th.47">GROUP_6:47</a></i>;<br/>


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