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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">g</font> =  <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#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>
<a NAME="E1:93"/>
 <a href="group_6.html#K10" title="GROUP_6:func.10">Ker</a> <span class="p1">(<span class="default"><a href="group_6.html#K9" title="GROUP_6:func.9">nat_hom</a> <span class="p2">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T52" target="_self" title="GROUP_6:th.52">Th52</a></span>;<br/>
<a NAME="E2:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A1</font></span>: 
 <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#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T66" target="_self" title="GROUP_6:th.66">Th66</a></span>;<br/>
<a NAME="E3:93"/>
 <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#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T69" target="_self" title="GROUP_6:th.69">Th69</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:93"/>
 <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#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c1">G</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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