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

<b>let </b><font color="Maroon">M</font> be   <a href="midsp_1.html#NM1">MidSp</a>;<br/>

<b>set </b><font color="Maroon">GS</font> =  <a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font>;<br/>
<b>thus </b><a NAME="E1:73"/>
 <a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font> is <a href="rlvect_1.html#V4">add-associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a><br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font>, <font color="Maroon">yy</font> = <font color="Maroon">y</font>, <font color="Maroon">zz</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<font color="Maroon">yy</font></span>)</span>,<font color="Maroon">zz</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">yy</font></span>)</span>,<font color="Maroon">zz</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">yy</font></span>)</span> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">zz</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<span class="p1">(<span class="default"><font color="Maroon">yy</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">xx</font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">yy</font>,<font color="Maroon">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>

;<br/>


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

<b>thus </b><a NAME="E2:73"/>
 <a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font> is <a href="rlvect_1.html#V5">right_zeroed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> ;<br/>
<b>thus </b><a NAME="E2:73_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">xxx</font> = <font color="Maroon">xx</font> as    <a href="midsp_1.html#M1" target="_self">Vector</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="midsp_1.html#T24" target="_self">Th24</a></i>;<br/>
<i><font color="Green">E37</font></i>: <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<span class="p1">(<span class="default"><a href="midsp_1.html#K17" target="_self">zerovect</a> <font color="Maroon">M</font></span>)</span>

<br/>.= 
<font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K17" target="_self">zerovect</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
;<br/>
<font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K17" target="_self">zerovect</a> <font color="Maroon">M</font></span>)</span> = 
<font color="Maroon">xxx</font> <a href="midsp_1.html#K10" target="_self">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K9" target="_self">ID</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T25" target="_self">Th25</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D5" target="_self">Def5</a></i>
;<br/>
<b>hence </b><a NAME="E4:73_2_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


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


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

<b>thus </b><a NAME="E3:73"/>
 <a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font> is <a href="rlvect_1.html#V6">right_complementable</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><a href="midsp_1.html#K16" target="_self">complvect</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_1.html#K16" target="_self">complvect</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font></span>)</span>

<br/>.= 
<font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_1.html#K16" target="_self">complvect</a> <font color="Maroon">M</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>
<b>by </b><i/>
;<br/>


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

<b>thus </b><a NAME="E4:73"/>
 <a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font> is <a href="rlvect_1.html#V3">Abelian</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font>, <font color="Maroon">yy</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="midsp_1.html#K13" target="_self">setvect</a> <font color="Maroon">M</font> ;<br/>
<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K15" target="_self">addvect</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">xx</font>,<font color="Maroon">yy</font>

<br/>.= 
<font color="Maroon">xx</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">yy</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">yy</font> <a href="midsp_1.html#K14" target="_self">+</a> <font color="Maroon">xx</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T26" target="_self">Th26</a></i>
<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K18" target="_self">vectgroup</a> <font color="Maroon">M</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T28" target="_self">Th28</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>

;<br/>


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


</div>
