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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> )</span><br/>

<b>set </b><font color="Maroon" title="c2">GS</font> =  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font>;<br/>
<b>thus </b><a NAME="E1:71"/>
 <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">xx</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">yy</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">zz</font> = <font color="Maroon" title="c5">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="midsp_1.html#K12" title="MIDSP_1:func.12">setvect</a> <font color="Maroon" title="c1">M</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font> = 
<span class="p1">(<span class="default"><a href="midsp_1.html#K14" title="MIDSP_1:func.14">addvect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c7">yy</font></span>)</span>,<font color="Maroon" title="c8">zz</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c7">yy</font></span>)</span> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c8">zz</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c6">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">yy</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c8">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#T75" target="_self" title="MIDSP_1:th.75">Th75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="midsp_1.html#K14" title="MIDSP_1:func.14">addvect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">xx</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">yy</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c8">zz</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:71"/>
 <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">xx</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="midsp_1.html#K12" title="MIDSP_1:func.12">setvect</a> <font color="Maroon" title="c1">M</font> ;<br/>
<b>thus </b><a NAME="E2:71_2"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c5">xxx</font> = <font color="Maroon" title="c4">xx</font> as    <a href="midsp_1.html#M1" title="MIDSP_1:mode.1">Vector</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="midsp_1.html#T71" target="_self" title="MIDSP_1:th.71">Th71</a></i>;<br/>
<font color="Maroon" title="c4">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K16" title="MIDSP_1:func.16">zerovect</a> <font color="Maroon" title="c1">M</font></span>)</span> = 
<font color="Maroon" title="c5">xxx</font> <a href="midsp_1.html#K9" title="MIDSP_1:func.9">+</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K8" title="MIDSP_1:func.8">ID</a> <font color="Maroon" title="c1">M</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#D14" target="_self" title="MIDSP_1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T64" target="_self" title="MIDSP_1:th.64">Th64</a></i>
;<br/>
<b>hence </b><a NAME="E3:71_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


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

<b>thus </b><a NAME="E3:71"/>
 <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c4">xx</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="midsp_1.html#K12" title="MIDSP_1:func.12">setvect</a> <font color="Maroon" title="c1">M</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">w</font> = <span class="p1">(<span class="default"><a href="midsp_1.html#K15" title="MIDSP_1:func.15">complvect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">xx</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">w</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span></span><br/>

<b>thus </b><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> = 
<font color="Maroon" title="c4">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="midsp_1.html#K15" title="MIDSP_1:func.15">complvect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">xx</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_1.html#D16" target="_self" title="MIDSP_1:def.16">Def16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E4:71"/>
 <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">xx</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">yy</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="midsp_1.html#K12" title="MIDSP_1:func.12">setvect</a> <font color="Maroon" title="c1">M</font> ;<br/>
<b>thus </b><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> = 
<font color="Maroon" title="c5">xx</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c6">yy</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c6">yy</font> <a href="midsp_1.html#K13" title="MIDSP_1:func.13">+</a> <font color="Maroon" title="c5">xx</font>
<b>by </b><i><a class="ref" href="midsp_1.html#T74" target="_self" title="MIDSP_1:th.74">Th74</a></i>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="midsp_1.html#D15" target="_self" title="MIDSP_1:def.15">Def15</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
