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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:30"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 holds <br/> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:30_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:30_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E3:30_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E4:30_1"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T82">FINSEQ_5:82</a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_7.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E6:30_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="txt" href="matrix_7.html#E3:30_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="matrix_7.html#E5:30_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_4.html#T11">GROUP_4:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E3:30"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<a NAME="E2:30_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 holds <br/> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:30_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<a NAME="E2:30_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="matrix_7.html#E2:30_2_1"><i><font color="Green">E29</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1_1"><b>case </b></a><a NAME="E1:30_2_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:30_2_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:30_2_1_1_1_1"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:30_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1_1_2"><b>case </b></a><a NAME="E1:30_2_1_1_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:30_2_1_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:30_2_1_1_1_2"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E2:30_2_1_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E5:30_2_1_1_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T24">FINSEQ_5:24</a>, <a class="txt" href="matrix_7.html#E1:30_2_1_1_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E6:30_2_1_1_1_2"/><b>then </b>
 <a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T66">FINSEQ_5:66</a></i>;<br/><a NAME="E7:30_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrix_7.html#T22" target="_self">Th22</a></i>;<br/><b>then </b><i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a>, <a class="txt" href="matrix_7.html#E5:30_2_1_1_1_2"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>
;<br/><i><font color="Green">E35</font></i>:  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_7.html#T23" target="_self">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
;<br/> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E8:30_2_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p3">(<span class="default"><span class="p4"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p4">(<span class="default"><span class="p5"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix_7.html#E9:30_2_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
;<br/><b>hence </b><a NAME="E11:30_2_1_1_1_2"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E1:30_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="matrix_7.html#E4:30_2_1_1_1_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_2_1"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:30"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="matrix_7.html#E2:30"><i><font color="Green">E26</font></i></a>, <a class="txt" href="matrix_7.html#E3:30"><i><font color="Green">E27</font></i></a>);<br/></i>
<a NAME="E5:30"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>]
 
;<br/>
<b>hence </b><a NAME="E6:30"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
