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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> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/>



<b>set </b><font color="Maroon" title="c3">g</font> =  <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">G'</font> = <font color="Maroon" title="c2">G</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">H</font> =  <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 <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span> #) as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E20"><i><font color="Green" title="E19">Lm7</font></i></a></i>;<br/>
<a NAME="E3:131"/><i><font color="Green" title="E107">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">H</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"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c4">G'</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></i>;<br/>
<i><font color="Green" title="E108">A2</font></i>:  <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span> = 
 <a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <font color="Maroon" title="c5">H</font>
<b>by </b><i><a class="ref" href="group_9.html#D23" target="_self" title="GROUP_9:def.23">Def23</a></i>
<br/>.= 
 <a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p1">(<span class="default"><a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c4">G'</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E3:131"><i><font color="Green" title="E107">A1</font></i></a>, <a class="ref" href="group_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></i>
;<br/>
<a NAME="E5:131"/>
 <a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p1">(<span class="default"><a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c4">G'</font></span>)</span> <a href="group_6.html#NR3" title="GROUP_6:NR.3">is_isomorphism</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T75" title="GROUP_6:th.75">GROUP_6:75</a></i>;<br/>
<a NAME="E6:131"/><b>then </b>
 <a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p1">(<span class="default"><a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c4">G'</font></span>)</span> <a href="group_6.html#NR1" title="GROUP_6:NR.1">is_monomorphism</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#D14" title="GROUP_6:def.14">GROUP_6:def 14</a></i>;<br/>
<a NAME="E7:131"/><b>then </b>
 <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="group_9.html#E4:131"><i><font color="Green" title="E108">A2</font></i></a>, <a class="ref" href="group_6.html#D12" title="GROUP_6:def.12">GROUP_6:def 12</a></i>;<br/>
<a NAME="E8:131"/><b>then </b>
(  <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_9.html#NR2" title="GROUP_9:NR.2">is_monomorphism</a>  &amp;  <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_9.html#NR3" title="GROUP_9:NR.3">is_epimorphism</a>  )
 
<b>by </b><i><a class="ref" href="group_9.html#D19" target="_self" title="GROUP_9:def.19">Def19</a>, <a class="ref" href="group_9.html#T53" target="_self" title="GROUP_9:th.53">Th53</a></i>;<br/>
<b>hence </b><a NAME="E9:131"/>
 <a href="group_9.html#K12" title="GROUP_9:func.12">nat_hom</a> <span class="p1">(<span class="default"><a href="group_9.html#K4" title="GROUP_9:func.4">(1).</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_9.html#NR4" title="GROUP_9:NR.4">is_isomorphism</a> 
 <b>by </b><i><a class="ref" href="group_9.html#D21" target="_self" title="GROUP_9:def.21">Def21</a></i>;<br/>


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