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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>] <b>means </b>for <font color="Olive" title="b1">F</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b2">b</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; ( <font color="Olive" title="b2">b</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  or (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Olive" title="b1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Olive" title="b1">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> $1</span>)</span>;<br/>
<a NAME="E1:59"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">F</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c6">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/>



<b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E40">A2</font></i>: 
( <font color="Maroon" title="c6">b</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; ( <font color="Maroon" title="c6">b</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  or (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) )
 ;<br/>

<b>thus </b><font color="Maroon" title="c6">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c6">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c5">F</font>
<b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:59_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></i>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K4" title="AFINSQ_1:func.4">&lt;%&gt;</a> <font color="Maroon" title="c1">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:59_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="stirl2_1.html#D3" target="_self" title="STIRL2_1:def.3">Def3</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:59"/><i><font color="Green" title="E40">A3</font></i>: 
for <font color="Olive" title="b1">G</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font><br/> for <font color="Olive" title="b2">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p1"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Olive" title="b2">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">G</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c6">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>;<br/>



<b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c7">F</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D</font>;<br/>



<b>assume </b><a NAME="E2:59_2"/><i><font color="Green" title="E42">A5</font></i>: 
( <font color="Maroon" title="c8">b</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; ( <font color="Maroon" title="c8">b</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  or (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_2_1_1"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1"><b>suppose </b></a><a NAME="E1:59_2_1_1_1"/><i><font color="Green" title="E43">A6</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_1"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:59_2_1_1_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:59_2_1_1_1"><i><font color="Green" title="E43">A6</font></i></a></i>;<br/>
<a NAME="E2:59_2_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">b</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  or (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:59_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E3:59_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="stirl2_1.html#T45" target="_self" title="STIRL2_1:th.45">Th45</a></i>;<br/>
<b>hence </b><a NAME="E4:59_2_1_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:59_2_1_1_1"/><i><font color="Green" title="E45">A8</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:59_2_1_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>then </b><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
<b>by </b><i><a class="txt" href="stirl2_1.html#E1:59_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="stirl2_1.html#E2:59_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="stirl2_1.html#E2:59_2_1_1_1"><i><font color="Green" title="E44">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c5">G</font></span>)</span>,<font color="Maroon" title="c6">d</font></span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:59_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E3:59_2_1_1_1"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="stirl2_1.html#T45" target="_self" title="STIRL2_1:th.45">Th45</a></i>
;<br/><b>hence </b><a NAME="E5:59_2_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_2"><b>suppose </b></a><a NAME="E1:59_2_1_1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:59_2_1_1_2"/><b>then </b><i><font color="Green" title="E43">A9</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><b>hence </b><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">d</font>
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:59_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="stirl2_1.html#T45" target="_self" title="STIRL2_1:th.45">Th45</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="stirl2_1.html#T44" target="_self" title="STIRL2_1:th.44">Th44</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="stirl2_1.html#E2:59_2_1_1_2"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:59_2"/>
<font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <span class="p3"><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K9" title="AFINSQ_1:func.9">%&gt;</a></span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:59"/>
for <font color="Olive" title="b1">G</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c1">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>]
 
<b>from </b><i><a class="ref" href="stirl2_1.html#S5" target="_self" title="STIRL2_1:sch.5">STIRL2_1:sch 5</a>(<a class="txt" href="stirl2_1.html#E1:59"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="stirl2_1.html#E2:59"><i><font color="Green" title="E40">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:59"/>
( <font color="Maroon" title="c4">b</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; ( <font color="Maroon" title="c4">b</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  or (  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c3">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) implies <font color="Maroon" title="c4">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">^</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="stirl2_1.html#K4" title="STIRL2_1:func.4">"**"</a> <font color="Maroon" title="c3">G</font></span>)</span> )
 ;<br/>


</div>
