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

<b>let </b><font color="Maroon">V</font> be   <a href="clvect_1.html#NM2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">H</font>, <font color="Olive">I</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">I</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">k</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="Olive">H</font></span>)</span> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> ) holds <br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">I</font></span>)</span>;<br/>
<a NAME="E1:24"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:24_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E2:24_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">k</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">H</font></span>)</span> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font>
 ;<br/><a NAME="E3:24_1_1"/>
(  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> &amp;  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:24_1_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:24"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:24_2_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">H</font>, <font color="Olive">I</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">I</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">k</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="Olive">H</font></span>)</span> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Olive">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> ) holds <br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">I</font></span>)</span>
 ;<br/><b>let </b><font color="Maroon">H</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>, <font color="Maroon">I</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:24_2_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E3:24_2_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E4:24_2_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">k</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">H</font></span>)</span> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font>
 ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">H</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">q</font> = <font color="Maroon">I</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E6:24_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E7:24_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1_1"><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>let </b><font color="Maroon">v</font> be   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:24_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</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">p</font></span>)</span>
 <b>and </b><br/><a NAME="E2:24_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E3:24_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E4:24_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
 <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">p</font></span>)</span> <a href="tarski.html#R1">c=</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">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E5:24_2_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E6:24_2_1_1"/><b>then </b>
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E7:24_2_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E8:24_2_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><b>hence </b><a NAME="E9:24_2_1_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div><a NAME="E9:24_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">n</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><b>then reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">v2</font> = <font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> <b>by </b><i>, , <a class="ref" href="rlvect_1.html#T54">RLVECT_1:54</a></i>;<br/><a NAME="E11:24_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v2</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E12:24_2_1"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</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">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:24_2_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</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">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">H</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v1</font>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#T55">RLVECT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i>, , , , <a class="ref" href="rlvect_1.html#T55">RLVECT_1:55</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:24"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</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">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:24"/>
for <font color="Olive">n</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">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:24"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">v</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <font color="Olive">v</font> ) implies  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="clvect_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i/>;<br/>


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