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

<b>let </b><font color="Maroon">X</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>, <font color="Maroon">Y</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/>

<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_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font><br/> for <font color="Olive">y</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font><br/> for <font color="Olive">z</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></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">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) holds <br/> <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E1:12"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:12_1"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 <b>and </b><br/><a NAME="E2:12_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/><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:12_1"/>
( <font color="Maroon">x</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">X</font> &amp; <font color="Maroon">y</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">Y</font> &amp; <font color="Maroon">z</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 <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b>
(  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> &amp;  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">Y</font> &amp;  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E5:12_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:12"/><i><font color="Green">E41</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="12_2"><b>proof </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="E1:12_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>thus </b><a NAME="E2:12_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="convfun1.html#K3" target="_self">Prod_of_RLS</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:12_2_1"/><i><font color="Green">E46</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</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="E2:12_2_1"/><i><font color="Green">E47</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:12_2_1"/><i><font color="Green">E48</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/>( <font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_1"><b>proof </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="E1:12_2_1_1"/>
<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:12_2_1_1"/><b>then </b>
( 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">n</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_1_1"/>
( <font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:12_2_1"/><i><font color="Green">E70</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/>
<a NAME="E5:12_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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> &amp; <font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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> &amp; <font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:12_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:12_2_1"/><i><font color="Green">E72</font></i>: 
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:12_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <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">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E9:12_2_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E10:12_2_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> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2"><b>proof </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="E1:12_2_1_2"/><i><font color="Green">E75</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">n</font>
 ;<br/>

<a NAME="E2:12_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:12_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:12_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:12_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E13:12_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E14:12_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E77</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
;<br/>
<a NAME="E16:12_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E17:12_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E18:12_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><i><font color="Green">E78</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
;<br/>
<a NAME="E20:12_2_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</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> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E21:12_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E22:12_2_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then </b> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</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="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
;<br/>
<b>hence </b><a NAME="E24:12_2_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:12"/>
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>(, );</i><br/>


</div>
