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

<b>assume </b><a NAME="E1:84"/><i><font color="Green">E55</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="setwop_2.html#K3">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <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="finseqop.html#K4">|-&gt;</a> 0;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <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="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:84"/><i><font color="Green">E56</font></i>: 
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">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/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:84_1"/><b>then </b>
<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="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:84_1"/>
<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="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="rvsum_1.html#E1:84"><i><font color="Green">E55</font></i></a>, <a class="txt" href="rvsum_1.html#E3:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rvsum_1.html#E1:84_1"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div>
<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E5:84"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:84"/><i><font color="Green">E59</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E7:84"/>
<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="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rvsum_1.html#E5:84"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E8:84"/><b>then </b>
 <a href="rvsum_1.html#K15" target="_self">Sum</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="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rvsum_1.html#E3:84"><i><font color="Green">E56</font></i></a>, <a class="txt" href="rvsum_1.html#E4:84"><i><font color="Green">E57</font></i></a>, <a class="txt" href="rvsum_1.html#E5:84"><i><font color="Green">E58</font></i></a>, <a class="txt" href="rvsum_1.html#E6:84"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rvsum_1.html#T113" target="_self">Th113</a></i>;<br/>
<b>hence </b><a NAME="E9:84"/>
0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="rvsum_1.html#K15" target="_self">Sum</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T111" target="_self">Th111</a></i>;<br/>


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