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

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>,  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>) <b>-&gt; </b>  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> = $1 <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c4">op</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span><b> such that </b><br/><a NAME="E2:97_1_1"/><i><font color="Green" title="E64">A1</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> holds  <font color="Maroon" title="c4">op</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon" title="c5">ch</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #);<br/>
<a NAME="E3:97_1_1"/><i><font color="Green" title="E65">A2</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #) is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</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="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">A'</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c10">B'</font> = <font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c11">C'</font> = <font color="Maroon" title="c8">C</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> ;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">C</font> = 
<font color="Maroon" title="c4">op</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B'</font></span>)</span>,<font color="Maroon" title="c8">C</font>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B'</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c11">C'</font>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c11">C'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T91" title="XBOOLE_1:th.91">XBOOLE_1:91</a></i>
<br/>.= 
<font color="Maroon" title="c4">op</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">B'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c11">C'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></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>
<a NAME="E4:97_1_1"/><i><font color="Green" title="E66">A3</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #) is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</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="c6">A</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> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">A'</font> = <font color="Maroon" title="c6">A</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> ;<br/>
<b>thus </b><font color="Maroon" title="c6">A</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> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #)</span>)</span> = 
<font color="Maroon" title="c7">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">A</font>

; <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>
<a NAME="E5:97_1_1"/><i><font color="Green" title="E67">A4</font></i>: 
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #) is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</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="c6">A</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="c7">A'</font> = <font color="Maroon" title="c6">A</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">A</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="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #)</span><br/>

<b>thus </b><font color="Maroon" title="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">A</font> = 
<font color="Maroon" title="c7">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c7">A'</font>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #)
<b>by </b><i><a class="ref" href="xboole_1.html#T92" title="XBOOLE_1:th.92">XBOOLE_1:92</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>
<a NAME="E6:97_1_1"/>
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #) is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c7">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</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="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">A</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">A'</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c9">B'</font> = <font color="Maroon" title="c7">B</font> as   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> ;<br/>
<b>thus </b><font color="Maroon" title="c6">A</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">B</font> = 
<font color="Maroon" title="c8">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c9">B'</font>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">B</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">A</font>
<b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></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>then reconsider </b><font color="Maroon" title="c6">ch</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# <span class="p1">(<span class="default"><a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">op</font>,<span class="p1">(<span class="default"><a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> #) as  <a href="algstr_0.html#V8" title="ALGSTR_0:attr.8">strict</a> <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> <b>by </b><i><a class="txt" href="#E3:97_1_1"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="#E4:97_1_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="#E5:97_1_1"><i><font color="Green" title="E67">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">ch</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">ch</font><br/> for <font color="Olive" title="b3">A'</font>, <font color="Olive" title="b4">B'</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B'</font> holds <br/><font color="Olive" title="b1">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">B'</font> ) )</span><br/>

<b>thus </b><a NAME="E8:97_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K1" title="CHAIN_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="chain_1.html#K4" title="CHAIN_1:func.4">cells</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font> &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">ch</font><br/> for <font color="Olive" title="b3">A'</font>, <font color="Olive" title="b4">B'</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B'</font> holds <br/><font color="Olive" title="b1">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">B'</font> ) )</span><br/>

<b>thus </b><a NAME="E9:97_1_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">ch</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="chain_1.html#K11" title="CHAIN_1:func.11">0_</a> <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">ch</font><br/> for <font color="Olive" title="b3">A'</font>, <font color="Olive" title="b4">B'</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B'</font> holds <br/><font color="Olive" title="b1">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">B'</font></span><br/>

<b>let </b><font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c8">B</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">ch</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A'</font>, <font color="Olive" title="b2">B'</font> being  <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>  st <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B'</font> holds <br/><font color="Maroon" title="c7">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b2">B'</font></span><br/><b>let </b><font color="Maroon" title="c9">A'</font>, <font color="Maroon" title="c10">B'</font> be   <a href="chain_1.html#NM4" title="CHAIN_1:NM.4">Chain</a> of <font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A'</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B'</font> implies <font color="Maroon" title="c7">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B'</font> )</span><br/>



<b>assume </b><a NAME="E10:97_1_1"/>
( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A'</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B'</font></span><br/>

<b>hence </b><a NAME="E11:97_1_1"/>
<font color="Maroon" title="c7">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A'</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B'</font>
 <b>by </b><i><a class="txt" href="#E2:97_1_1"><i><font color="Green" title="E64">A1</font></i></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>
