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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:15"/><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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="extreal1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E3:15"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="extreal1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15_1_1"/><i><font color="Green">E18</font></i>: 
0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:15_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="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>6</sub></font> =  <a href="supinf_2.html#K1">0.</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E4:15_1_1"/><i><font color="Green">E19</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E5:15_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="mesfunc3.html#E4:15_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:15"/><i><font color="Green">E18</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="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:15_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>

<a NAME="E2:15_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="extreal1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15_2_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <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>5</sub></font>
 <b>and </b><br/><a NAME="E2:15_2_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E3:15_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:15_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63">FINSEQ_1:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K6">ExtREAL</a>  <b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a></i>;<br/>
<i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E9:15_2_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 
;<br/>
<a NAME="E10:15_2_1"/>
 <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<a NAME="E11:15_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:15_2_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E13:15_2_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mesfunc3.html#E8:15_2_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mesfunc3.html#E11:15_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:15_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mesfunc3.html#E13:15_2_1"><i><font color="Green">E26</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<a NAME="E16:15_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_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="E1:15_2_1_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:15_2_1_3"/><b>then </b><i><font color="Green">E28</font></i>: 
( 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>9</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:15_2_1_3"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:15_2_1_3"/><b>then </b><i><font color="Green">E29</font></i>: 
( 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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:15_2_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="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></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E6:15_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E4:15_2_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:15_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:15_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:15_2_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mesfunc3.html#E1:15_2_1_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E18:15_2_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="mesfunc3.html#E9:15_2_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E20:15_2_1"/><i><font color="Green">E28</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</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> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E22:15_2_1"/><i><font color="Green">E29</font></i>: 
(  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="ref" href="convfun1.html#D5">CONVFUN1:def 5</a></i>;<br/>
<a NAME="E23:15_2_1"/><i><font color="Green">E30</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="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_4"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:15_2_1_4"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E20:15_2_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="mesfunc3.html#E22:15_2_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:15_2_1_4"/><i><font color="Green">E32</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</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="15_2_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:15_2_1_4_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>15</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2_1_4_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:15_2_1_4_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:15_2_1_4_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1_4_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:15_2_1_4_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E22:15_2_1"><i><font color="Green">E29</font></i></a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="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></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E6:15_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E3:15_2_1_4_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:15_2_1_4_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E20:15_2_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E8:15_2_1_4_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:15_2_1_4_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1_4_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E10:15_2_1_4_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1]
 <b>by </b><i><a class="txt" href="mesfunc3.html#E5:15_2_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mesfunc3.html#E1:15_2_1_4_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mesfunc3.html#E2:15_2_1_4_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="mesfunc3.html#E4:15_2_1_4_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="mesfunc3.html#E7:15_2_1_4_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2_1_4_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>15</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:15_2_1_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</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="mesfunc3.html#E1:15_2_1_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="mesfunc3.html#E2:15_2_1_4"><i><font color="Green">E32</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:15_2_1_4"/>
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="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E24:15_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then </b> <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E1:15_2_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="mesfunc3.html#E20:15_2_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E9:15_2_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="mesfunc3.html#E23:15_2_1"><i><font color="Green">E30</font></i></a></i>
;<br/>
<b>then </b><i><font color="Green">E31</font></i>:  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mesfunc3.html#E2:15_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mesfunc3.html#E13:15_2_1"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="mesfunc3.html#E8:15_2_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="mesfunc3.html#E22:15_2_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K10">1.</a>  <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mesfunc3.html#E18:15_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="extreal2.html#T4">EXTREAL2:4</a></i>
;<br/>
<b>take </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="supinf_2.html#K2">+</a> <a href="mesfunc1.html#K10">1.</a> ;<br/>

<a NAME="E27:15_2_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_2_1_7"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="mesfunc3.html#E18:15_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mesfunc1.html#D8">MESFUNC1:def 8</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2_1_7_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_7_1_1_1"><b>case </b></a><a NAME="E1:15_2_1_7_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_2_1_7_1_1_1"/>
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E18:15_2_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="mesfunc3.html#E26:15_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="extreal1.html#T29">EXTREAL1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1_7_1_1_2"><b>case </b></a><a NAME="E1:15_2_1_7_1_1_2"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:15_2_1_7_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E18:15_2_1"><i><font color="Green">E27</font></i></a></i>;<br/><a NAME="E3:15_2_1_7_1_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="mesfunc1.html#K10">1.</a> 
 <b>by </b><i><a class="ref" href="mesfunc1.html#D8">MESFUNC1:def 8</a></i>;<br/><b>hence </b><a NAME="E4:15_2_1_7_1_1_2"/>
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E26:15_2_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="mesfunc3.html#E2:15_2_1_7_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="extreal1.html#T29">EXTREAL1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2_1_7"/>
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E28:15_2_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/>
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="mesfunc3.html#E3:15"><i><font color="Green">E17</font></i></a>, <a class="txt" href="mesfunc3.html#E4:15"><i><font color="Green">E18</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> <b> such that </b><br/><a NAME="E7:15"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="mesfunc3.html#E2:15"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>thus </b><a NAME="E8:15"/>
 <a href="convfun1.html#K5">Sum</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="extreal1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="mesfunc3.html#E1:15"><i><font color="Green">E15</font></i></a>, <a class="txt" href="mesfunc3.html#E7:15"><i><font color="Green">E19</font></i></a></i>;<br/>


</div>
