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<b>let </b><font color="Maroon">ADG</font> be   <a href="afvect0.html#NM2">Proper_Uniquely_Two_Divisible_Group</a>;<br/><b>let </b><font color="Maroon">o'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:79"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 ;<br/>

<b>set </b><font color="Maroon">AS</font> =  <a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font>;<br/>
<b>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>;<br/>
<b>set </b><font color="Maroon">Y</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">Z</font> =  <a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font>;<br/>
<b>set </b><font color="Maroon">T</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> = <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="binop_1.html#NM1">UnOp</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font><b> such that </b><br/><a NAME="E3:79"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E4:79"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="tdgroup.html#T9">TDGROUP:9</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">g</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span> ;<br/>
<a NAME="E6:79"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E7:79"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T26" target="_self">Th26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font>;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> = 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a>, <a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T24" target="_self">Th24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>

;<br/><b>thus </b><a NAME="E2:79_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">ADG</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p2">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font></span>)</span>
 <b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a>, <a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T24" target="_self">Th24</a></i>;<br/><b>reconsider </b><font color="Maroon">fa</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span> ;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="afvect0.html#K4" target="_self">Pcom</a> <font color="Maroon">o</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">fa</font>
<b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a>, <a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T24" target="_self">Th24</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#D2" target="_self">Def2</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E9:79"/><b>then </b>
<font color="Maroon">f</font> <a href="afvect0.html#R3" target="_self">is_Iso_of</a> <font color="Maroon">ADG</font>, <a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T29" target="_self">Th29</a>, <a class="ref" href="afvect0.html#T30" target="_self">Th30</a>, <a class="ref" href="afvect0.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E10:79"/>
<font color="Maroon">ADG</font>, <a href="afvect0.html#K5" target="_self">GroupVect</a> <span class="p1">(<span class="default"><a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font></span>)</span>,<font color="Maroon">o</font> <a href="afvect0.html#R4" target="_self">are_Iso</a> 
 <b>by </b><i><a class="ref" href="afvect0.html#T23" target="_self">Th23</a></i>;<br/>


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