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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">a<sub>1</sub></font>  st ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">x</font> ) );<br/>
<a NAME="E1:16"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> 0;<br/>

<b>assume </b><a NAME="E1:16_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 ;<br/>

<a NAME="E2:16_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#T2">EUCLID:2</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 0 holds <br/><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> ) )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 ;<br/><b>thus </b><a NAME="E2:16_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">xk</font> = <font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font>;<br/>
<a NAME="E1:16_2_1_1"/>
<font color="Maroon">x</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<a NAME="E2:16_2_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</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">x</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:16_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:16_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E5:16_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">k</font> <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/>
<b>then reconsider </b><font color="Maroon">xk</font> = <font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">k</font> <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:16_2_1_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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="E3:16_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E4:16_2_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:16_2_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E6:16_2_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:16_2_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">xk</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:16_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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="E10:16_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <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">x</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:16_2_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <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">x</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:16_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">xk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<i><font color="Green">E74</font></i>: <font color="Maroon">x</font> = 
<font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>
<br/>.= 
<font color="Maroon">xk</font> <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">x</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T11">FINSEQ_5:11</a></i>
;<br/>
<a NAME="E14:16_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <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"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E15:16_2_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:16_2_1_1"/><i><font color="Green">E80</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">xk</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_2_1_1_3"/>
<font color="Maroon">i</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">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/><a NAME="E2:16_2_1_1_3"/><b>then </b><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1_3_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_3_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_3_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:16_2_1_1_3_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:16_2_1_1_3_1_1"/><b>then </b>
<font color="Maroon">xk</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">xk</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:16_2_1_1_3_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">xk</font>
 <b>by </b><i>, <a class="txt" href="real_ns1.html#E1"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E6:16_2_1_1_3_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:16_2_1_1_3_1_1"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">xk</font> <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">xk</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T33">XREAL_1:33</a></i>;<br/><b>hence </b><a NAME="E8:16_2_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_3_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_1_1_3_1_2"/>
<font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T33">XREAL_1:33</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E18:16_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</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">k</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x</font> ) )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T35">XREAL_1:35</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:16_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:16"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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