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

<span class="kw">reconsider </span><font color="Maroon" title="c2">A0</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>] <span class="kw">means </span> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> );<br/>
<a NAME="E2:17_1_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
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="c1">X</font>  ex <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">c</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c4">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">c</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:17_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</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>
<span class="kw">consider </span><font color="Maroon" title="c3">h</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E4:17_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
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="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c3">h</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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c4">G</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #);<br/>
<a NAME="E5:17_1_1"/><span class="lab"><font color="Green" title="E7">A3</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</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="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</span></span><br/>

<a NAME="E1:17_1_1_2"/>
 ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y9</font> &amp; <font color="Maroon" title="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x9</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y9</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:17_1_1_2"/>
<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">y</font></span>)</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="E6:17_1_1"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> &amp; <font color="Olive" title="b3">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <font color="Olive" title="b2">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:17_1_1_3"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">b</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">b</font></span>)</span> = 
<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span> <a href="bcialg_1.html#K2" title="BCIALG_1:func.2">`</a> </span>)</span>

<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T54" title="BCIALG_1:th.54">BCIALG_1:54</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#D5" title="BCIALG_1:def.5">BCIALG_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:17_1_1_3"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1_3"><span class="lab"><font color="Green" title="E9">A5</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="E7:17_1_1"/>
(  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) 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="17_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:17_1_1_4"/>
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #); <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="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font> = 
<font color="Maroon" title="c7">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T57" title="BCIALG_1:th.57">BCIALG_1:57</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</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>

<div><span class="kw">hereby </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"> (  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #); <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="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c7">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font> = 
<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span>,<font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T57" title="BCIALG_1:th.57">BCIALG_1:57</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T7" title="BCIALG_1:th.7">BCIALG_1:7</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T57" title="BCIALG_1:th.57">BCIALG_1:57</a></span>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c10">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</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>

<div><span class="kw">hereby </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">  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</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="c5">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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><span class="kw">thus </span><font color="Maroon" title="c5">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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #)</span>)</span> = 
<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T2" title="BCIALG_1:th.2">BCIALG_1:2</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bcialg_1.html#T2" title="BCIALG_1:th.2">BCIALG_1:2</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">let </span><font color="Maroon" title="c5">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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #); <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="c5">a</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E5:17_1_1_4"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">b</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_1"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</font> as   <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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">b</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="c5">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>  <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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #)</span><br/>

<span class="kw">thus </span><a NAME="E7:17_1_1_4"/>
<font color="Maroon" title="c5">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>  <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>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a>, <a class="txt" href="#E5:17_1_1_4"><span class="lab"><font color="Green" title="E9">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><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">G</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c2">A0</font> #) 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> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">G</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="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">thus </span><a NAME="E9:17_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="bcialg_1.html#K1" title="BCIALG_1:func.1">\</a> <font color="Olive" title="b2">y</font></span>)</span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_1"><span class="lab"><font color="Green" title="E7">A3</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>
