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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4" target="_self">add-associative</a> <a href="rlvect_1.html#V5" target="_self">right_zeroed</a>  <a href="rlvect_1.html#L1" target="_self">LoopStr</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">G</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">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Olive">G</font></span>)</span>;<br/>
<a NAME="E1:61"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><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>assume </b><a NAME="E1:61_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:61_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:61_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp;  <a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1" target="_self">0.</a> <font color="Maroon">V</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1"/>
 <a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:61"/><i><font color="Green">E32</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>proof </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="E1:61_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">G</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">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> holds <br/> <a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Olive">G</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>;<br/>

<b>assume </b><a NAME="E2:61_2"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">H</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</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="E4:61_2"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:61_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</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">H</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E6:61_2"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> 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#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E8:61_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</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="E9:61_2"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E10:61_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E11:61_2"/>
 <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">k</font>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E12:61_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_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="E1:61_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:61_2_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">H</font> &amp; <font color="Maroon">n</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="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="rlvect_1.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><b>hence </b><a NAME="E3:61_2_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_2"><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="E1:61_2_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E2:61_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E3:61_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:61_2_2"/>
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:61_2"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D8" target="_self">Def8</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E16:61_2"/><b>then </b>
<font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E17:61_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></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="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E18:61_2"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></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="E19:61_2"/><i><font color="Green">E94</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E95</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E21:61_2"/><i><font color="Green">E96</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E22:61_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <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">F</font></span>)</span> <a href="nat_1.html#K1">+</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="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E23:61_2"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" target="_self">Th10</a>, <a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E24:61_2"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rlvect_1.html#E18"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_2_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><i><font color="Green">E101</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_4_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:61_2_4_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:61_2_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_4_2"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:61_2_4_2"/><b>then </b><i><font color="Green">E102</font></i>: 
not <font color="Maroon">n</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">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="E3:61_2_4_2"/><i><font color="Green">E103</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E18"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:61_2_4_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:61_2_4_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><div><i><font color="Green">E105</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_4_2_1"/>
not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:61_2_4_2_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:61_2_4_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16" target="_self">Th16</a>, <a class="ref" href="rlvect_1.html#T19" target="_self">Th19</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2_4_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_2_4_2_2"/>
not <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><a NAME="E2:61_2_4_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</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">F</font></span>)</span> <a href="nat_1.html#K1">+</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="txt" href="rlvect_1.html#E18"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="rlvect_1.html#D10" target="_self">Def10</a>, , <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E3:61_2_4_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/></div><b>end;</b></div><a NAME="E9:61_2_4_2"/><b>then </b>
<font color="Maroon">j</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="rlvect_1.html#T23" target="_self">Th23</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:61_2_4_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D5" target="_self">Def5</a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E11:61_2_4_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" target="_self">Def7</a>, <a class="ref" href="rlvect_1.html#T21" target="_self">Th21</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E12:61_2_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D8" target="_self">Def8</a>, <a class="ref" href="rlvect_1.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:61_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="rlvect_1.html#E23"><i><font color="Green">Lemma56</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E107</font></i>: <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E27:61_2"/><i><font color="Green">E108</font></i>: 
 <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/>
<font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<b>hence </b> <a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" target="_self">Def9</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">v</font>
<b>by </b><i><a class="txt" href="rlvect_1.html#E4:61_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="txt" href="rlvect_1.html#E5:61_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D6" target="_self">Def6</a>, <a class="txt" href="rlvect_1.html#E23"><i><font color="Green">Lemma56</font></i></a>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:61"/>
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>(, );<br/></i>
<a NAME="E4:61"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> implies  <a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">G</font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E5:61"/>
 <a href="rlvect_1.html#K9" target="_self">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="rlvect_1.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9" target="_self">Sum</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>


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