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

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">R1</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">R2</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">R1</font>, <font color="Olive">R2</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">a<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>   st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">R1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">R2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> ) holds <br/> <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Olive">R1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Olive">R2</font>;<br/>
<a NAME="E1:81"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R1</font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">R2</font> be    <a href="finseq_2.html#M2">Element</a> of 0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:81_1"/>
(  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_2.html#T113">FINSEQ_2:113</a></i>;<br/>
<b>hence </b><a NAME="E2:81_1"/>
( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 0 holds <br/><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> ) implies  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:81"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:81_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">R1</font>, <font color="Olive">R2</font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>   st ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> holds <br/><font color="Olive">R1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">R2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> ) holds <br/> <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Olive">R1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Olive">R2</font>
 ;<br/>

<b>let </b><font color="Maroon">R1</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">R2</font> be    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E2:81_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font>
 ;<br/>

<b>consider </b><font color="Maroon">R1'</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:81_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x1</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>consider </b><font color="Maroon">R2'</font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:81_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2'</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x2</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T137">FINSEQ_2:137</a></i>;<br/>
<b>set </b><font color="Maroon">n</font> = <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<a NAME="E7:81_2"/>
( <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">R2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T20" target="_self">Th20</a>, <a class="ref" href="rvsum_1.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a>, <a class="ref" href="finseq_2.html#T136">FINSEQ_2:136</a></i>;<br/>
<a NAME="E8:81_2"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E24"><i><font color="Green">Lemma55</font></i></a></i>;<br/>
<a NAME="E9:81_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">R1'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2'</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:81_2_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E2:81_2_1"/>
(  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R1'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R1'</font> &amp;  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">R2'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">R2'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:81_2_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">R1'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">R2'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T20" target="_self">Th20</a>, <a class="ref" href="rvsum_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rvsum_1.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a>, <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/>
<b>hence </b><a NAME="E4:81_2_1"/>
<font color="Maroon">R1'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2'</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E24"><i><font color="Green">Lemma55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:81_2"/><b>then </b>
(  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2'</font> &amp;  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1'</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">x1</font> &amp;  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2'</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T20" target="_self">Th20</a>, <a class="ref" href="rvsum_1.html#T21" target="_self">Th21</a>, <a class="ref" href="rvsum_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E11:81_2"/>
 <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T22" target="_self">Th22</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:81"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="rvsum_1.html#T16" target="_self">Th16</a>, <a class="ref" href="rvsum_1.html#T17" target="_self">Th17</a>);<br/></i>
<b>hence </b><a NAME="E4:81"/>
( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">R1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> ) implies  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">R2</font> )
 ;<br/>


</div>
