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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<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="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) holds <br/>(  <a href="wsierp_1.html#K8">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1 );<br/>
<a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:2_1"/><i><font color="Green">E4</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_1_1"/>
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> = 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="uproots.html#E1:2_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E1:2_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1
 ;<br/>

<b>thus </b><a NAME="E5:2_1"/>
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:2_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:2"/><i><font color="Green">E4</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="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:2_2"/><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>and </b><br/><a NAME="E3:2_2"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:2_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<a NAME="E6:2_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T27">FINSEQ_2:27</a></i>;<br/>
<a NAME="E7:2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:2_2"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E9:2_2"/><b>then </b><i><font color="Green">E11</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>2</sub></font>
 
;<br/>
<div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:2_2_1"/><i><font color="Green">E14</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><a NAME="E3:2_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uproots.html#E1:2_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:2_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E3:2_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="uproots.html#E1:2_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="uproots.html#E2:2_2_1"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:2_2"/><i><font color="Green">E13</font></i>: 
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uproots.html#E6:2_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<a NAME="E12:2_2"/><i><font color="Green">E14</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E13:2_2"/><i><font color="Green">E15</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uproots.html#E8:2_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E15:2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="uproots.html#E3:2_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="uproots.html#E12:2_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="uproots.html#E13:2_2"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E16:2_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E17:2_2"/><b>then </b><i><font color="Green">E16</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:2_2_2"/><i><font color="Green">E17</font></i>: 
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1 as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E3:2_2_2"/><i><font color="Green">E18</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</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="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/><a NAME="E4:2_2_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E5:2_2_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="txt" href="uproots.html#E6:2_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:2_2_2_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:2_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:2_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="uproots.html#E10:2_2"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E4:2_2_2_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E5:2_2_2_1"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</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="E6:2_2_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:2_2_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/></div><b>end;</b></div><a NAME="E7:2_2_2"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E3:2_2_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="uproots.html#E4:2_2_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="uproots.html#E5:2_2_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/><i><font color="Green">E22</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> 1
<b>by </b><i><a class="txt" href="uproots.html#E9:2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>

;<br/><i><font color="Green">E23</font></i>:  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uproots.html#E6:2_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="uproots.html#E1:2_2_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
;<br/><a NAME="E10:2_2_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E7:2_2_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="uproots.html#E8:2_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E11:2_2_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:2_2_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E17:2_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E13:2_2_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E1:2_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="uproots.html#E9:2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="uproots.html#E10:2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="uproots.html#E9:2_2_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E14:2_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uproots.html#E8:2_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="uproots.html#E12:2_2_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_2.html#T74">FINSEQ_2:74</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E19:2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1
 ;<br/>

<b>then </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default">1 <a href="finsop_1.html#K1">|-&gt;</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="finseq_2.html#T143">FINSEQ_2:143</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finsop_1.html#K1">|-&gt;</a> 1</span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a></i>
;<br/>
<a NAME="E21:2_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="uproots.html#E5:2_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="uproots.html#E9:2_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_2.html#T20">FINSEQ_2:20</a></i>;<br/>
<b>hence </b><a NAME="E22:2_2"/>
 <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:2_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="uproots.html#E2:2_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="uproots.html#E9:2_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="uproots.html#E10:2_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="uproots.html#E11:2_2"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:2"/>
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="uproots.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="uproots.html#E2:2"><i><font color="Green">E4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:2"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) implies (  <a href="wsierp_1.html#K8">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finsop_1.html#K1">|-&gt;</a> 1 ) )
 ;<br/>


</div>
