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

<span class="kw">deffunc </span><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>) <span class="kw">-&gt; </span>  <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/>
<span class="kw">consider </span><font color="Maroon" title="c4">op</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <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><span class="kw"> such that </span><br/><a NAME="E2:99_1_1"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span>
<span class="kw">set </span><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="E3:99_1_1"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 <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="99_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">A9</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c10">B9</font> = <font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c11">C9</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/>
<span class="kw">thus </span><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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B9</font></span>)</span>,<font color="Maroon" title="c8">C</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B9</font></span>)</span> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c11">C9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c11">C9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T91" title="XBOOLE_1:th.91">XBOOLE_1:91</a></span>
<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">B9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c11">C9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:99_1_1"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <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="99_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">A9</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/>
<span class="kw">thus </span><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">A9</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">A</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:99_1_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
 <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="99_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">A9</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/>
<span class="kw">take </span>
<font color="Maroon" title="c6">A</font>
; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c7">A9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
<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> #)
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T92" title="XBOOLE_1:th.92">XBOOLE_1:92</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:99_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="99_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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> #); <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">A9</font> = <font color="Maroon" title="c6">A</font>, <font color="Maroon" title="c9">B9</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/>
<span class="kw">thus </span><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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c9">B9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:99_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E4:99_1_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E5:99_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">ch</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font>, <font color="Olive" title="b4">B9</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">A9</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B9</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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">B9</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:99_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font>, <font color="Olive" title="b4">B9</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">A9</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B9</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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">B9</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:99_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font>, <font color="Olive" title="b4">B9</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">A9</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B9</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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b4">B9</font></span><br/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A9</font>, <font color="Olive" title="b2">B9</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">A9</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B9</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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Olive" title="b2">B9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">A9</font>, <font color="Maroon" title="c10">B9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font> &amp; <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B9</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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B9</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E10:99_1_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A9</font>
 <span class="kw">and </span><br/><a NAME="E11:99_1_1"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B9</font></span><br/>

<span class="kw">thus </span><a NAME="E12:99_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">A9</font> <a href="chain_1.html#K7" title="CHAIN_1:func.7">+</a> <font color="Maroon" title="c10">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:99_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E10:99_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E11:99_1_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
