<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font> holds   <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="Olive" title="b1">G</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="group_9.html#E20"><span class="lab"><font color="Green" title="E19">Lm7</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">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/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">G9</font> = <font color="Maroon" title="c2">G</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<a NAME="E3:127"/><span class="lab"><font color="Green" title="E104">A1</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c5">G9</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D8" target="_self" title="GROUP_9:def.8">Def8</a></span>;<br/>
<a NAME="E4:127"/><span class="lab"><font color="Green" title="E105">A2</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="c5">G9</font></span>)</span> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</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> 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_9.html#T53" target="_self" title="GROUP_9:th.53">Th53</a>, <a class="ref" href="group_6.html#T75" title="GROUP_6:th.75">GROUP_6:75</a></span>;<br/>
 <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="c3">H</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_9.html#D23" target="_self" title="GROUP_9:def.23">Def23</a></span>
<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#K6" title="GROUP_2:func.6">(1).</a> <font color="Maroon" title="c5">G9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:127"><span class="lab"><font color="Green" title="E104">A1</font></span></a>, <a class="ref" href="group_2.html#D7" title="GROUP_2:def.7">GROUP_2:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:127"/>
 <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_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:127"><span class="lab"><font color="Green" title="E105">A2</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/>


</div>
