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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K4" target="_self">.--&gt;</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K4" target="_self">.--&gt;</a> the <a href="struct_0.html#U2">Zero</a> of <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/>
<a NAME="E2:20_1_1"/>
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> )
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fvsum_1.html#T2">FVSUM_1:2</a>, <a class="ref" href="fvsum_1.html#T3">FVSUM_1:3</a></i>;<br/>
<a NAME="E3:20_1_1"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp;  <a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a> )
 
<b>by </b><i><a class="ref" href="prvect_1.html#T14" target="_self">Th14</a>, <a class="ref" href="prvect_1.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E4:20_1_1"/><i><font color="Green">E17</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="prvect_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K4" target="_self">.--&gt;</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a>  <a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E6:20_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</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>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:20_1_1_1"/><i><font color="Green">E19</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="binop_1.html#V2">associative</a> &amp;  <a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="txt" href="prvect_1.html#E5"><i><font color="Green">Lemma3</font></i></a>, <a class="ref" href="fvsum_1.html#T3">FVSUM_1:3</a></i>;<br/>
<a NAME="E2:20_1_1_1"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 
<b>by </b><i><a class="txt" href="prvect_1.html#E4:20_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/>
<a NAME="E3:20_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="prvect_1.html#K3" target="_self">product</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="finseqop.html#R1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="prvect_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="prvect_1.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:20_1_1_1"/>
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green">E15</font></i></a>, <a class="ref" href="prvect_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:20_1_1_1"/><b>then </b><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E7:20_1_1_1"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" target="_self">product</a> <span class="p3">(<span class="default"><a href="vectsp_1.html#K7">comp</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E3:20_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseqop.html#D1">FINSEQOP:def 1</a></i>;<br/>
<a NAME="E8:20_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>;<br/>
<a NAME="E9:20_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E6:20_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<a NAME="E10:20_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>hence </b><a NAME="E11:20_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1_1"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" target="_self">product</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prvect_1.html#K4" target="_self">.--&gt;</a> the <a href="struct_0.html#U2">Zero</a> of <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#E3:20_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="prvect_1.html#E5:20_1_1"><i><font color="Green">E18</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/>


</div>
