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

<b>let </b><font color="Maroon">GF</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">GF</font>;<br/><b>let </b><font color="Maroon">L</font> be    <a href="vectsp_6.html#M1">Linear_Combination</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">F</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="relat_1.html#K2">rng</a> <font color="Olive">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">K</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> st  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Olive">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">K</font>;<br/>
<a NAME="E1:7"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<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>;<br/>

<b>assume </b><a NAME="E1:7_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon">F</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:7_1"/><b>then </b>
<font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</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>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T33">VECTSP_6:33</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K2">ZeroLC</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T41">VECTSP_6:41</a></i>
;<br/>
<a NAME="E5:7_1"/>
 <a href="vectsp_6.html#K2">ZeroLC</a> <font color="Maroon">V</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T26">VECTSP_6:26</a></i>;<br/>
<b>hence </b><a NAME="E6:7_1"/>
 ex <font color="Olive">K</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> st  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">K</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">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="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:7_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:7_2"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E3:7_2"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/>

<a NAME="E4:7_2"/>
<font color="Maroon">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:7_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T21">FINSEQ_2:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font>, <font color="Maroon">x'</font> = <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></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#T50">FINSEQ_1:50</a></i>;<br/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K7">^</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/>
<a NAME="E9:7_2"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:7_2"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:7_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span> )
 
;<br/>
<a NAME="E12:7_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">LA1</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E14:7_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA1</font>
 <b>by </b><i><a class="ref" href="vectsp_7.html#T12">VECTSP_7:12</a></i>;<br/>
<a NAME="E15:7_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="vectsp_4.html#T17">VECTSP_4:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></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">x</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/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></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="E18:7_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">LA2</font> being    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E20:7_2"/><i><font color="Green">E63</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA2</font>
 <b>by </b><i>, </i>;<br/>
<i><font color="Green">E65</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">x'</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="vectsp_6.html#T37">VECTSP_6:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA2</font></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">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">x'</font></span>)</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="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA2</font></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="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T34">VECTSP_6:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p2">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <font color="Maroon">LA2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">LA1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T78">VECTSP_6:78</a></i>
<br/>.= 
 <a href="vectsp_6.html#K4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">LA2</font> <a href="vectsp_6.html#K5">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">LA1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_6.html#T77">VECTSP_6:77</a></i>
;<br/>
<a NAME="E22:7_2"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">LA1</font> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T61">VECTSP_6:61</a></i>;<br/>
<a NAME="E23:7_2"/><b>then </b>
<font color="Maroon">LA2</font> <a href="vectsp_6.html#K5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="vectsp_6.html#K6">*</a> <font color="Maroon">LA1</font></span>)</span> is    <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_6.html#T52">VECTSP_6:52</a></i>;<br/>
<b>hence </b><a NAME="E24:7_2"/>
 ex <font color="Olive">K</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> st  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">K</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:7"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">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#S2">NAT_1:sch 2</a>(, );<br/></i>
<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>;<br/>

<b>assume </b><a NAME="E4:7"/><i><font color="Green">E70</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E5:7"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> is   <a href="nat_1.html#NM1">Nat</a>
 
;<br/>
<b>hence </b><a NAME="E6:7"/>
 ex <font color="Olive">K</font> being   <a href="vectsp_6.html#M2">Linear_Combination</a> of <font color="Maroon">A</font> st  <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="vectsp_6.html#K3">(#)</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="vectsp_6.html#K4">Sum</a> <font color="Olive">K</font>
 <b>by </b><i>, </i>;<br/>


</div>
