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

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">.--&gt;</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">F</font></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon" title="c4">G</font> =  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">.--&gt;</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">F</font></span>)</span> #) as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">LoopStr</a>  ;<br/>
<a NAME="E2:20_1_1"/>
( the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="fvsum_1.html#T2" title="FVSUM_1:th.2">FVSUM_1:2</a>, <a class="ref" href="fvsum_1.html#T3" title="FVSUM_1:th.3">FVSUM_1:3</a></i>;<br/>
<a NAME="E3:20_1_1"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
(  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp;  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 
<b>by </b><i><a class="ref" href="prvect_1.html#T14" target="_self" title="PRVECT_1:th.14">Th14</a>, <a class="ref" href="prvect_1.html#T15" target="_self" title="PRVECT_1:th.15">Th15</a></i>;<br/>
<a NAME="E4:20_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">F</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="prvect_1.html#T3" target="_self" title="PRVECT_1:th.3">Th3</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">.--&gt;</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">F</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T16" target="_self" title="PRVECT_1:th.16">Th16</a></i>;<br/>
<a NAME="E6:20_1_1"/>
<font color="Maroon" title="c4">G</font> is <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">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" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a><br/>

<b>set </b><font color="Maroon" title="c6">C</font> =  <a href="vectsp_1.html#K7" title="VECTSP_1:func.7">comp</a> <font color="Maroon" title="c1">F</font>;<br/>
<b>set </b><font color="Maroon" title="c7">B</font> = the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>;<br/>
<a NAME="E1:20_1_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
( the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp;  <a href="vectsp_1.html#K7" title="VECTSP_1:func.7">comp</a> <font color="Maroon" title="c1">F</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font> )
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="prvect_1.html#E5"><i><font color="Green" title="E3">Lm1</font></i></a>, <a class="ref" href="fvsum_1.html#T3" title="FVSUM_1:th.3">FVSUM_1:3</a></i>;<br/>
<a NAME="E2:20_1_1_1"/>
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 
<b>by </b><i><a class="txt" href="prvect_1.html#E4:20_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/>
<a NAME="E3:20_1_1_1"/><b>then </b><i><font color="Green" title="E20">A6</font></i>: 
 <a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <span class="p1">(<span class="default"><a href="vectsp_1.html#K7" title="VECTSP_1:func.7">comp</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c2">n</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a>  <a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="prvect_1.html#T17" target="_self" title="PRVECT_1:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">y</font> = <span class="p1">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <span class="p2">(<span class="default"><a href="vectsp_1.html#K7" title="VECTSP_1:func.7">comp</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:20_1_1_1"/>
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">F</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E1:20"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="prvect_1.html#T3" target="_self" title="PRVECT_1:th.3">Th3</a></i>;<br/>
<a NAME="E6:20_1_1_1"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c4">G</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="ref" href="prvect_1.html#T16" target="_self" title="PRVECT_1:th.16">Th16</a></i>;<br/>
<a NAME="E7:20_1_1_1"/>
<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prvect_1.html#K3" title="PRVECT_1:func.3">product</a> <span class="p3">(<span class="default"><a href="vectsp_1.html#K7" title="VECTSP_1:func.7">comp</a> <font color="Maroon" title="c1">F</font></span>)</span>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E3:20_1_1_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></i>;<br/>
<a NAME="E8:20_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
;<br/>
<a NAME="E9:20_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="prvect_1.html#E6:20_1_1_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>
<a NAME="E10:20_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">G</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" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">G</font> st <font color="Maroon" title="c5">x</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">+</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c4">G</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:20_1_1"/>
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="prvect_1.html#K2" title="PRVECT_1:func.2">product</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prvect_1.html#K4" title="PRVECT_1:func.4">.--&gt;</a> the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">F</font></span>)</span> #) is  <a href="rlvect_1.html#V1" title="RLVECT_1:attr.1">strict</a> <a href="vectsp_1.html#NM1" title="VECTSP_1:NM.1">AbGroup</a>
 <b>by </b><i><a class="txt" href="prvect_1.html#E3:20_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="prvect_1.html#E5:20_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="prvect_1.html#E7"><i><font color="Green" title="E4">Lm2</font></i></a>, <a class="txt" href="prvect_1.html#E8"><i><font color="Green" title="E5">Lm3</font></i></a></i>;<br/>


</div>
