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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>] <b>means </b>for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><br/> for <font color="Olive">g</font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>  st <font color="Olive">g</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Olive">g</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or (  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) ) holds <br/><font color="Olive">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Olive">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Olive">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:14"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">g</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">g</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) ) )
 ;<br/>

<a NAME="E2:14_1"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<b>thus </b><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span></span>)</span> = 
<font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setwiseo.html#T23">SETWISEO:23</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:14"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">G</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><br/> for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E1:14_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">G</font>]
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/>



<b>assume </b><a NAME="E2:14_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">g</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">g</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) ) )
 ;<br/>

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_2_1"/>
not <font color="Maroon">g</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 ;<br/><a NAME="E2:14_2_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E3:14_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green">E43</font></i>: <font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">d</font>
<b>by </b><i>, </i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_3_1_1"><b>suppose </b></a><a NAME="E1:14_2_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:14_2_3_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<font color="Maroon">d</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">G</font></span>)</span>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="ref" href="binop_1.html#D3">BINOP_1:def 3</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_3_1_2"><b>suppose </b></a><a NAME="E1:14_2_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:14_2_3_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> = 
<font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">d</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:14_2"/>
<font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:14"/>
for <font color="Olive">G</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">G</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:14"/>
( <font color="Maroon">g</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">g</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  or (  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 ) ) implies <font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finsop_1.html#K2" target="_self">"**"</a> <font color="Maroon">G</font></span>)</span> )
 ;<br/>


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