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

<span class="kw">let </span><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> ; <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">F</font>, <font color="Olive" title="b2">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="b3">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="b3">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b3">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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> <font color="Olive" title="b2">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b3">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Olive" title="b1">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Olive" title="b2">G</font></span>)</span></span><br/><span class="kw">let </span><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>; <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">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="b1">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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 ) ) holds <br/><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c2">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c3">G</font></span>)</span></span><br/><span class="kw">let </span><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>; <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="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> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c3">G</font></span>)</span> )</span><br/>







<span class="kw">defpred </span><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>] <span class="kw">means </span>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> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> $1</span>)</span>;<br/>
<a NAME="E1:81"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
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="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Olive" title="b2">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <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">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="Maroon" title="c5">G</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Olive" title="b1">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>]</span><br/><span class="kw">let </span><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>; <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="c5">G</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:81_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>]</span><br/>

<span class="kw">let </span><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>; <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">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="b1">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) holds <br/><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span></span><br/><span class="kw">let </span><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>; <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="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> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) ) implies <font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:81_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c8">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">and </span><br/><a NAME="E3:81_1"/><span class="lab"><font color="Green" title="E49">A4</font></span>: 
( <font color="Maroon" title="c8">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 ) )
 ; <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="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:81_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="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="81_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:81_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A5</font></span>: 
 <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> 
 ; <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="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:81_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="ordinal4.html#K1" title="ORDINAL4:func.1">^</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>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1"><span class="lab"><font color="Green" title="E49">A4</font></span></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></span>;<br/><a NAME="E3:81_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T52" target="_self" title="AFINSQ_2:th.52">Th53</a></span>;<br/><a NAME="E4:81_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></span>;<br/><a NAME="E5:81_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#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E2:81_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="txt" href="#E3:81_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="txt" href="#E4:81_1_1_1_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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#K4" title="BINOP_1:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c5">G</font></span>)</span>,<font color="Maroon" title="c6">d</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:81_1_1_1_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="afinsq_2.html#T52" target="_self" title="AFINSQ_2:th.52">Th53</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:81_1_1_1_1"/>
<font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:81_1_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> 
 ; <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="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:81_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<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> 
 ;<br/><span class="kw">hence </span><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c7">F</font></span>)</span>,<font color="Maroon" title="c6">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:81_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="afinsq_2.html#T52" target="_self" title="AFINSQ_2:th.52">Th53</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T49" target="_self" title="AFINSQ_2:th.49">Th50</a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_1_1_1_2"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></span>
;<br/> <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">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:81_1"/>
<font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p3"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c6">d</font></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span></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="E2:81"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<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="81_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <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">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="b1">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Olive" title="b1">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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 ) ) holds <br/><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c5">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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></span><br/><span class="kw">let </span><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>; <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">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  &amp; ( <font color="Maroon" title="c6">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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 ) ) implies <font color="Maroon" title="c6">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:81_2"/>
<font color="Maroon" title="c6">b</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
 <span class="kw">and </span><br/><a NAME="E2:81_2"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
( <font color="Maroon" title="c6">b</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="finseq_1.html#K3" title="FINSEQ_1:func.3">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 ) )
 ; <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">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c6">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c5">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_2"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></span>
<br/>.= 
<font color="Maroon" title="c6">b</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:81_2"><span class="lab"><font color="Green" title="E48">A9</font></span></a>, <a class="ref" href="afinsq_2.html#D9" target="_self" title="AFINSQ_2:def.9">Def9</a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</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="E3:81"/>
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="afinsq_2.html#S2" target="_self" title="AFINSQ_2:sch.2">AFINSQ_2:sch 2</a>(<a class="txt" href="#E2:81"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E1:81"><span class="lab"><font color="Green" title="E46">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:81"/>
( <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> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</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#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</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="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c3">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"> verum</span><br/>


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