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

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font>,  <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font>) <b>-&gt; </b>  <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> = <font color="Maroon">a<sub>1</sub></font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">op</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="chain_1.html#K1" target="_self">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span><b> such that </b><br/><a NAME="E2:102_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> holds  <font color="Maroon">op</font> <a href="binop_1.html#K2">.</a> <font color="Olive">A</font>,<font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">A</font>,<font color="Olive">B</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon">ch</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #);<br/>
<a NAME="E3:102_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #) is <a href="rlvect_1.html#V4">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #), <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #), <font color="Maroon">C</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</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">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font>, <font color="Maroon">C'</font> = <font color="Maroon">C</font> as   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">B</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">C</font> = 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">B'</font></span>)</span>,<font color="Maroon">C</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">B'</font></span>)</span> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T91">XBOOLE_1:91</a></i>
<br/>.= 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">C'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:102_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #) is <a href="rlvect_1.html#V5">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</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">A'</font> = <font color="Maroon">A</font> as   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> ;<br/>
<b>thus </b><font color="Maroon">A</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #)</span>)</span> = 
<font color="Maroon">op</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A'</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E5:102_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #) is <a href="rlvect_1.html#V6">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</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">A'</font> = <font color="Maroon">A</font> as   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> ;<br/>
<b>take </b>
<font color="Maroon">A</font>
;<br/>

<b>thus </b><font color="Maroon">A</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">A</font> = 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">A'</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #)
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E6:102_1_1"/>
 <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #) is <a href="rlvect_1.html#V3">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #), <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</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">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font> as   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font> ;<br/>
<b>thus </b><font color="Maroon">A</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">B</font> = 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">B'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">op</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">B</font>,<font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">B</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">ch</font> =  <a href="rlvect_1.html#G1">LoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" target="_self">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span> #) as  <a href="rlvect_1.html#V1">strict</a> <a href="vectsp_1.html#NM1">AbGroup</a> <b>by </b><i><a class="ref" href="chain_1.html#T7" target="_self">Th7</a>, <a class="ref" href="chain_1.html#T8" target="_self">Th8</a>, <a class="ref" href="chain_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>take </b>
<font color="Maroon">ch</font>
;<br/>

<b>thus </b><a NAME="E8:102_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">ch</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K1" target="_self">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K5" target="_self">cells</a> <font color="Maroon">k</font>,<font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E9:102_1_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">ch</font> <a href="hidden.html#R1">=</a>  <a href="chain_1.html#K12" target="_self">0_</a> <font color="Maroon">k</font>,<font color="Maroon">G</font>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ch</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">ch</font>;<br/><b>let </b><font color="Maroon">A'</font> be   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font>, <font color="Maroon">B'</font> be   <a href="chain_1.html#NM4">Chain</a> of <font color="Maroon">k</font>,<font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E10:102_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> )
 ;<br/>

<b>hence </b><font color="Maroon">A</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">B</font> = 
<font color="Maroon">op</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">A'</font>,<font color="Maroon">B'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T5">RLVECT_1:5</a></i>
<br/>.= 
<font color="Maroon">A'</font> <a href="chain_1.html#K8" target="_self">+</a> <font color="Maroon">B'</font>
<b>by </b><i/>
;<br/><br/>


</div>
