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

<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">f</font> be   <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 <font color="Maroon">ADG</font>;<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><b>that </b><br/><a NAME="E1:76"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">x</font>
 <b>and </b><br/><a NAME="E2:76"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font>
 ;<br/>

<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>set </b><font color="Maroon">a'</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">b'</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>;<br/>
<a NAME="E3:76"/><i><font color="Green">E38</font></i>: 
 <a href="tdgroup.html#K2">AV</a> <font color="Maroon">ADG</font> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ADG</font>,<span class="p1">(<span class="default"><a href="tdgroup.html#K1">CONGRD</a> <font color="Maroon">ADG</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="tdgroup.html#D5">TDGROUP:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a''</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>, <font color="Maroon">b''</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</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><a NAME="E5:76"/>
<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afvect0.html#K3" target="_self">Padd</a> <font color="Maroon">o</font></span>)</span> <a href="binop_1.html#K1">.</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<a NAME="E1:76_1"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="afvect0.html#K3" target="_self">Padd</a> <font color="Maroon">o</font></span>)</span> <a href="binop_1.html#K1">.</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> <a href="hidden.html#R1">=</a>  <a href="afvect0.html#K2" target="_self">Padd</a> <font color="Maroon">o</font>,<font color="Maroon">a''</font>,<font color="Maroon">b''</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T5" target="_self">Th5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c''</font> = <span class="p1">(<span class="default"><a href="afvect0.html#K3" target="_self">Padd</a> <font color="Maroon">o</font></span>)</span> <a href="binop_1.html#K1">.</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> 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>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c''</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font> <b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E4:76_1"/>
<font color="Maroon">o</font>,<font color="Maroon">a''</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b''</font>,<font color="Maroon">c''</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:76_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o'</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">c'</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tdgroup.html#K1">CONGRD</a> <font color="Maroon">ADG</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T26" target="_self">Th26</a>, <a class="ref" href="afvect0.html#T27" target="_self">Th27</a>, <a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>
<a NAME="E6:76_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</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> <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>
 
<b>by </b><i><a class="ref" href="tdgroup.html#D4">TDGROUP:def 4</a></i>;<br/>
<a NAME="E7:76_1"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="afvect0.html#T25" target="_self">Th25</a></i>;<br/>
<b>then </b><font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c'</font> = 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">o'</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T29" target="_self">Th29</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<b>then </b><font color="Maroon">c'</font> = 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="ref" href="afvect0.html#T25" target="_self">Th25</a></i>
;<br/>
<b>hence </b><a NAME="E10:76_1"/>
<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afvect0.html#K3" target="_self">Padd</a> <font color="Maroon">o</font></span>)</span> <a href="binop_1.html#K1">.</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>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:76"/>
<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<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#K1">0.</a> <font color="Maroon">ADG</font></span>)</span> = 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">ADG</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T25" target="_self">Th25</a></i>
<br/>.= 
 <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#T26" target="_self">Th26</a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:76"/>
<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> <a href="hidden.html#R1">=</a> <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_1.html#K1">.</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>proof </b></a><div class="add">

<a NAME="E1:76_3"/><i><font color="Green">E42</font></i>: 
<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_1.html#K1">.</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> <a href="hidden.html#R1">=</a>  <a href="afvect0.html#NK3" target="_self">Pcom</a> <font color="Maroon">o</font>,<font color="Maroon">a''</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T6" target="_self">Th6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c''</font> = <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_1.html#K1">.</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> 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>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c''</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ADG</font> <b>by </b><i><a class="ref" href="afvect0.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E4:76_3"/>
<font color="Maroon">a''</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">c''</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T28" target="_self">Th28</a>, <a class="ref" href="afvect0.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:76_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">o'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">o'</font>,<font color="Maroon">c'</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="tdgroup.html#K1">CONGRD</a> <font color="Maroon">ADG</font>
 
<b>by </b><i><a class="ref" href="afvect0.html#T26" target="_self">Th26</a>, <a class="ref" href="afvect0.html#T27" target="_self">Th27</a>, <a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>
<a NAME="E6:76_3"/><b>then </b>
<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> <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">o'</font>
 
<b>by </b><i><a class="ref" href="tdgroup.html#D4">TDGROUP:def 4</a></i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>: <font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">o'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="afvect0.html#T25" target="_self">Th25</a></i>
<br/>.= 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">c'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<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> = 
<font color="Maroon">o'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">a</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="afvect0.html#T29" target="_self">Th29</a>, <a class="ref" href="rlvect_1.html#T21">RLVECT_1:21</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c'</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">ADG</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">c'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E9:76_3"/>
<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> <a href="hidden.html#R1">=</a> <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_1.html#K1">.</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>
 ;<br/>


</div><b>end;</b></div>


</div>
