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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="supinf_1.html#K3">REAL</a> , <a href="supinf_1.html#K3">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:21"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="supinf_1.html#K3">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E3:21"/><i><font color="Green">E16</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> ) implies <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:21_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E18</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:21_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>, <font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>, <font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T121">FINSEQ_2:121</a></i>;<br/>
<i><font color="Green">E19</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>
<br/>.= 
1

;<br/>
<a NAME="E5:21_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="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<i><font color="Green">E21</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
 <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>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> 2
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_2_2_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rfunct_4.html#E2:21_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_1_1"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:21_1_1_2_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_2_2_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:21_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rfunct_4.html#E4:21_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E7:21_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E9:21_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<b>then </b><i><font color="Green">E22</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#D9">RVSUM_1:def 9</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E11:21_1_1"/>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T87">RVSUM_1:87</a></i>;<br/>
<a NAME="E12:21_1_1"/><b>then </b>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E7:21_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E10:21_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E13:21_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E15:21_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<b>then </b><i><font color="Green">E24</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#D9">RVSUM_1:def 9</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>
<a NAME="E17:21_1_1"/>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T87">RVSUM_1:87</a></i>;<br/>
<a NAME="E18:21_1_1"/><b>then </b>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E7:21_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E16:21_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E19:21_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</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#K11">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E6:21_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rfunct_4.html#E13:21_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="txt" href="rfunct_4.html#E2:21"><i><font color="Green">E15</font></i></a>, <a class="ref" href="rfunct_3.html#D13">RFUNCT_3:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a>  <a href="supinf_1.html#K3">REAL</a>  implies for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>

<a NAME="E2:21_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>proof </b></a><div class="add">

<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>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></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="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_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="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>;<br/>
<a NAME="E1:21_2_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T109">RVSUM_1:109</a></i>;<br/>
<a NAME="E2:21_2_1"/><i><font color="Green">E19</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="21_2_1_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="E1:21_2_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_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="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:21_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <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="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_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="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <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="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:21_2_1_1_1"/><i><font color="Green">E21</font></i>: 
(  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( 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="finseq_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="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:21_2_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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">c<sub>8</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">c<sub>9</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:21_2_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></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">c<sub>10</sub></font> being    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:21_2_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></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/>
<a NAME="E8:21_2_1_1_1"/>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rfunct_4.html#E5:21_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rfunct_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:21_2_1_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<a NAME="E10:21_2_1_1_1"/>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rfunct_4.html#E7:21_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="rfunct_4.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E11:21_2_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 = 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
;<br/>
<a NAME="E13:21_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <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="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E14:21_2_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>11</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E7:21_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E14:21_2_1_1_1"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="rfunct_4.html#E5:21_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/>
<a NAME="E16:21_2_1_1_1"/><i><font color="Green">E29</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="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_2_1_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:21_2_1_1_1_3"/>
<font color="Maroon">c<sub>12</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>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:21_2_1_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:21_2_1_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:21_2_1_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:21_2_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:21_2_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</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="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E8:21_2_1_1_1_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:21_2_1_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:21_2_1_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E8:21_2_1_1_1_3"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1_1_1_4_1"/>
(  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_4_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_2_1_1_1_4_1_1"/><b>then </b>
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="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="rfunct_4.html#E16:21_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rfunct_4.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:21_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0
 <b>by </b><i><a class="ref" href="rfunct_4.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E4:21_2_1_1_1_4_1_1"/><b>then </b>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0
 <b>by </b><i><a class="ref" href="rfunct_4.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E5:21_2_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>;<br/><a NAME="E6:21_2_1_1_1_4_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_4_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/><a NAME="E7:21_2_1_1_1_4_1_1"/>
 <a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0
 <b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1_4_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rfunct_4.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:21_2_1_1_1_4_1_1"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K24">*</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>;<br/><b>hence </b><a NAME="E9:21_2_1_1_1_4_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E9:21_2_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="rfunct_4.html#E11:21_2_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rfunct_4.html#E15:21_2_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rfunct_4.html#E5:21_2_1_1_1_4_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="rfunct_4.html#E6:21_2_1_1_1_4_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_1_1_4_1_2"/><i><font color="Green">E30</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green">E31</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E9:21_2_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T94">RVSUM_1:94</a></i>
;<br/><a NAME="E3:21_2_1_1_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/><a NAME="E4:21_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E5:21_2_1_1_1_4_1_2"/><i><font color="Green">E33</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_4_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:21_2_1_1_1_4_1_2_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E16:21_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>

<a NAME="E2:21_2_1_1_1_4_1_2_2"/>
<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="binop_2.html#K23">+</a> 1 <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="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/>
<a NAME="E3:21_2_1_1_1_4_1_2_2"/><b>then </b>
<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="binop_2.html#K23">+</a> 1 <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="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E4:21_2_1_1_1_4_1_2_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<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="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:21_2_1_1_1_4_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E6:21_2_1_1_1_4_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E4:21_2_1_1_1_4_1_2_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:21_2_1_1_1_4_1_2_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="rfunct_4.html#E4:21_2_1_1_1_4_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:21_2_1_1_1_4_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="rfunct_4.html#E2:21_2_1_1_1_4_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="rfunct_4.html#E4:21_2_1_1_1_4_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rfunct_3.html#D13">RFUNCT_3:def 13</a></i>;<br/><a NAME="E7:21_2_1_1_1_4_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_4_1_2_3"><b>proof </b></a><div class="add">

<i><font color="Green">E35</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T94">RVSUM_1:94</a></i>
;<br/>
<i><font color="Green">E36</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_4_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
;<br/>
<a NAME="E3:21_2_1_1_1_4_1_2_3"/>
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="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1_1_4_1_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_2_1_1_1_4_1_2_3_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:21_2_1_1_1_4_1_2_3_3"/>
<font color="Maroon">c<sub>12</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:21_2_1_1_1_4_1_2_3_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:21_2_1_1_1_4_1_2_3_3"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T67">RVSUM_1:67</a></i>;<br/>
<a NAME="E5:21_2_1_1_1_4_1_2_3_3"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_4_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="rfunct_4.html#E5:21_2_1_1_1_4_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T124">XREAL_1:124</a></i>;<br/>
<a NAME="E6:21_2_1_1_1_4_1_2_3_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>12</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>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E7:21_2_1_1_1_4_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:21_2_1_1_1_4_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E16:21_2_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:21_2_1_1_1_4_1_2_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="rfunct_4.html#E4:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="rfunct_4.html#E5:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a></i>;<br/>

<a NAME="E10:21_2_1_1_1_4_1_2_3_3"/>
( <font color="Maroon">c<sub>12</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>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</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>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E11:21_2_1_1_1_4_1_2_3_3"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:21_2_1_1_1_4_1_2_3_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E6:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:21_2_1_1_1_4_1_2_3_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E14:21_2_1_1_1_4_1_2_3_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E15:21_2_1_1_1_4_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:21_2_1_1_1_4_1_2_3_3"/><b>then </b>
<font color="Maroon">c<sub>12</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="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E17:21_2_1_1_1_4_1_2_3_3"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:21_2_1_1_1_4_1_2_3_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E5:21_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="rfunct_4.html#E7:21_2_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="rfunct_4.html#E11:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E19:21_2_1_1_1_4_1_2_3_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfunct_4.html#E17:21_2_1_1_1_4_1_2_3_3"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_2_1_1_1_4_1_2_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:21_2_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_4_1_2_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="rfunct_4.html#E2:21_2_1_1_1_4_1_2_3"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:21_2_1_1_1_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E5:21_2_1_1_1_4_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E9:21_2_1_1_1_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E10:21_2_1_1_1_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E6:21_2_1_1_1_4_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E11:21_2_1_1_1_4_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E11:21_2_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="rfunct_4.html#E15:21_2_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="rfunct_4.html#E1:21_2_1_1_1_4_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:21_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_2_1_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="E3:21_2_1"/>
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="rfunct_4.html#E1:21_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="rfunct_4.html#E2:21_2_1"><i><font color="Green">E19</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:21_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:21"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <font color="Olive">b<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a>  st  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Olive">b<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) ) holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E3:21"><i><font color="Green">E16</font></i></a></i>;<br/>


</div>
