<?xml version="1.0"?>
<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>let </b><font color="Maroon">x</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:57"/><b>then </b><i><font color="Green">E56</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>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:57"/><i><font color="Green">E57</font></i>: 
 <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><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:57"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funcop_1.html#K8">[;]</a> <font color="Maroon">r</font>,<span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a></i>;<br/>
<b>set </b><font color="Maroon">h1</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font>;<br/>
<b>reconsider </b><font color="Maroon">h2</font> =  <a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a>  as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<a NAME="E6:57"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#D5">FUNCOP_1:def 5</a></i>;<br/>
<a NAME="E7:57"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> &amp; ( for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span> holds <br/><span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">u</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"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 
<b>by </b><i><a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<a NAME="E8:57"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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="E1:57_1"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
<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"><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#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:57_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:57_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E5:57_1"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E6:57_1"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E7:57_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<i><font color="Green">E67</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="numbers.html#K1">REAL</a> 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
;<br/>
<a NAME="E9:57_1"/><b>then </b><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p4">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">h2</font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <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"><span class="p4">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p5">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E4:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>;<br/>
<a NAME="E10:57_1"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> <a href="numbers.html#K1">REAL</a> </span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E3:57"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E11:57_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<a NAME="E12:57_1"/>
<font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K35">multreal</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan2c.html#E2:57"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E1:57_1"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="jordan2c.html#E4:57_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan2c.html#E4:57"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan2c.html#E7:57"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E13:57_1"/><b>then </b>
<font color="Maroon">g</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E6:57"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>;<br/>
<b>hence </b><a NAME="E14:57_1"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:57"/>
(  <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">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan2c.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>


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