<?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#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green">E48</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="relset_1.html#K4">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#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<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#K1">REAL</a>  st ( 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="relset_1.html#K4">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#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) &amp;  <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> holds <br/> <a href="rvsum_1.html#K16">Product</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0;<br/>
<a NAME="E2:49"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>;<br/>
<a NAME="E3:49"/><i><font color="Green">E50</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="49_1"><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="E1:49_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<a NAME="E2:49_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st ( 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="relset_1.html#K4">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#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) &amp;  <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">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 holds <br/> <a href="rvsum_1.html#K16">Product</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<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#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green">E52</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="relset_1.html#K4">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#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:49_1_1"/><i><font color="Green">E53</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
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:49_1_1"/><i><font color="Green">E54</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>
 <b>and </b><br/><a NAME="E5:49_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E6:49_1_1"/><i><font color="Green">E56</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="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:49_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_2.html#T26">FINSEQ_2:26</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K1">.</a> 1;<br/>
<a NAME="E7:49_1_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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>proof </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="E1:49_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:49_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T18">FINSEQ_2:18</a>, <a class="txt" href="nat_4.html#E6:49_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E3:49_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:49_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:49_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E1:49_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_4.html#E6:49_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:49_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:49_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nat_4.html#E4:49_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E9:49_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E10:49_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:49_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:49_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K1">.</a> <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="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:49_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="txt" href="nat_4.html#E6:49_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E12:49_1_1"/>
 <a href="finseq_1.html#K2">Seg</a> <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="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:49_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:49_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E14:49_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:49_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_4.html#E11:49_1_1"><i><font color="Green">E58</font></i></a></i>;<br/>
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="rvsum_1.html#K16">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"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E5:49_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="nat_4.html#E6:49_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K16">Product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
;<br/>
<b>hence </b><a NAME="E16:49_1_1"/>
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E14:49_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nat_4.html#E8:49_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:49"/><i><font color="Green">E51</font></i>: 
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="nat_4.html#E2:49"><i><font color="Green">E49</font></i></a>, <a class="txt" href="nat_4.html#E3:49"><i><font color="Green">E50</font></i></a>);<br/></i>
<a NAME="E5:49"/>
ex <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#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E6:49"/>
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="nat_4.html#E1:49"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nat_4.html#E4:49"><i><font color="Green">E51</font></i></a></i>;<br/>


</div>
