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

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" target="_self">zeros</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L1">LoopStr</a>  ;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/> <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="prvect_1.html#D12" target="_self">Def12</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E2:58_1_1_1"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>)
 <b>by </b><i><a class="ref" href="prvect_1.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:58_1_1_2"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) )
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:58_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="prvect_1.html#D13" target="_self">Def13</a></i>;<br/><b>hence </b><a NAME="E2:58_1_1_2"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T2">FVSUM_1:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:58_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span> is <a href="binop_1.html#V1">commutative</a>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T25" target="_self">Th25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:58_1_1_3"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) )
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:58_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="prvect_1.html#D13" target="_self">Def13</a></i>;<br/><b>hence </b><a NAME="E2:58_1_1_3"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="ref" href="fvsum_1.html#T3">FVSUM_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:58_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span> is <a href="binop_1.html#V2">associative</a>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T26" target="_self">Th26</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:58_1_1_4"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K20" target="_self">zeros</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>) )
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:58_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="prvect_1.html#D13" target="_self">Def13</a>, <a class="ref" href="prvect_1.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E2:58_1_1_4"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K20" target="_self">zeros</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K9" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="prvect_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:58_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="prvect_1.html#K20" target="_self">zeros</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E9:58_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <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">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="prvect_1.html#K12" target="_self">Frege</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K19" target="_self">complop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E1:58_1_1_5_1"/>
( <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>) <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>) &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K19" target="_self">complop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="prvect_1.html#K17" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="prvect_1.html#E2:58_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="prvect_1.html#D13" target="_self">Def13</a>, <a class="ref" href="prvect_1.html#D14" target="_self">Def14</a>, <a class="ref" href="prvect_1.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E2:58_1_1_5_1"/>
( <span class="p1">(<span class="default"><a href="prvect_1.html#K19" target="_self">complop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="prvect_1.html#K10" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 <b>by </b><i><a class="ref" href="prvect_1.html#T4" target="_self">Th4</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:58_1_1_5"/><b>then </b>
 <a href="prvect_1.html#K12" target="_self">Frege</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K19" target="_self">complop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E4:58_1_1_5"/><b>then </b>
<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>;<br/>
<a NAME="E5:58_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<a NAME="E6:58_1_1_5"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E8:58_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E7:58_1_1_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:58_1_1"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="prvect_1.html#K16" target="_self">carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<span class="p1"><a href="prvect_1.html#K14" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K18" target="_self">addop</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="prvect_1.html#K14" target="_self">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K20" target="_self">zeros</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is  <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a>
 <b>by </b><i><a class="txt" href="prvect_1.html#E4:58_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="prvect_1.html#E6:58_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="prvect_1.html#E8:58_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="prvect_1.html#E7"><i><font color="Green">Lemma4</font></i></a>, <a class="txt" href="prvect_1.html#E8"><i><font color="Green">Lemma5</font></i></a></i>;<br/>


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