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<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> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<b>consider </b><font color="Maroon">c<sub>3</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="E2:23"/><i><font color="Green">E16</font></i>: 
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>and </b><br/><a NAME="E4:23"/><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="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="convfun1.html#D5" target="_self">Def5</a></i>;<br/>
<i><font color="Green">E19</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
<b>by </b><i><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>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/>
<a NAME="E6:23"/>
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>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_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="E1:23_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:23_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E5:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:23_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:23"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E4:23_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:23_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="int_1.html#T20">INT_1:20</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E5:23_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</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>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:23_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:23_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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>1</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E3:23_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23"/><b>then </b><i><font color="Green">E20</font></i>: 
 <a href="convfun1.html#K5" target="_self">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="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E3:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="convfun1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E8:23"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E5:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:23"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</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>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="mesfunc1.html#K4">.</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>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:23"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:23"/>
 <a href="convfun1.html#K5" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="binarith.html#K13">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="convfun1.html#K5" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="convfun1.html#E2:23"><i><font color="Green">E16</font></i></a>, <a class="txt" href="convfun1.html#E5:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="convfun1.html#E7:23"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>


</div>
