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

<b>let </b><font color="Maroon" title="c1">ps</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">ps</font> holds <br/><font color="Maroon" title="c1">ps</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">ps1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:10"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c1">ps</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">ps1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></i>;<br/>
<a NAME="E4:10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">ps</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ps1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E5:10"/><b>then </b><i><font color="Green" title="E12">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">ps</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">ps1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:10"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E8:10"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">y2</font> = <font color="Maroon" title="c4">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:10"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c1">ps</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">ps</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="uniroots.html#E5:10"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E11:10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">ps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">ps</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E12:10"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">ps</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">ps</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c3">ps1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E14:10"/><i><font color="Green" title="E15">A7</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font> holds <br/><font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font>
 ;<br/>

<a NAME="E2:10_1"/><i><font color="Green" title="E17">A9</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">ps</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E3:10_1"/>
<font color="Maroon" title="c1">ps</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="uniroots.html#E1:10_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1"/>
<font color="Maroon" title="c6">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="uniroots.html#E1:10_1"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="uniroots.html#E2:10_1"><i><font color="Green" title="E17">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font> ) holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>;<br/>
<a NAME="E15:10"/><i><font color="Green" title="E16">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10_2"/><i><font color="Green" title="E17">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:10_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E3:10_2"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:10_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></i>;<br/>
<b>hence </b><a NAME="E4:10_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="uniroots.html#E10:10"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="uniroots.html#E12:10"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:10"/><i><font color="Green" title="E17">A12</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_3"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c8">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:10_3"/><i><font color="Green" title="E19">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>and </b><br/><a NAME="E3:10_3"/><i><font color="Green" title="E20">A15</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E4:10_3"/>
<font color="Maroon" title="c8">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:10_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">v</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c10">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:10_3"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">w</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T63" title="FINSEQ_1:th.63">FINSEQ_1:63</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f1</font> = <font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c12">f2</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">w</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="uniroots.html#E6:10_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<a NAME="E8:10_3"/>
<font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:10_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="uniroots.html#E6:10_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E9:10_3"/><b>then </b><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E10:10_3"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f1</font> holds <br/><font color="Maroon" title="c11">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:10_3_1"/><i><font color="Green" title="E23">A18</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f1</font>
 ;<br/>

<a NAME="E2:10_3_1"/><i><font color="Green" title="E24">A19</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E6:10_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E3:10_3_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E6:10_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="uniroots.html#E1:10_3_1"><i><font color="Green" title="E23">A18</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:10_3_1"/>
<font color="Maroon" title="c11">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:10_3"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="uniroots.html#E1:10_3_1"><i><font color="Green" title="E23">A18</font></i></a>, <a class="txt" href="uniroots.html#E2:10_3_1"><i><font color="Green" title="E24">A19</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:10_3"/><b>then </b><i><font color="Green" title="E23">A20</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:10_3"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="uniroots.html#E9:10_3"><i><font color="Green" title="E22">A17</font></i></a></i>;<br/>
<a NAME="E12:10_3"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
;<br/>
<a NAME="E13:10_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">w</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c10">w</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E15:10_3"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:10_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="uniroots.html#E6:10_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="uniroots.html#E9:10_3"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E16:10_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="uniroots.html#E2:10_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E17:10_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:10_3"/><b>then </b><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E3:10_3"><i><font color="Green" title="E20">A15</font></i></a>, <a class="txt" href="uniroots.html#E15:10_3"><i><font color="Green" title="E24">A21</font></i></a></i>;<br/>
<a NAME="E19:10_3"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:10_3"/><b>then </b><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="real_2.html#T144" title="REAL_2:th.144">REAL_2:144</a></i>;<br/>
<a NAME="E21:10_3"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E6:10_3"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></i>;<br/>
<a NAME="E22:10_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c11">f1</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">m</font>
 
;<br/>
<a NAME="E23:10_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">m</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E1:10"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="uniroots.html#E11:10_3"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="uniroots.html#E18:10_3"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<b>hence </b><a NAME="E24:10_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E20:10_3"><i><font color="Green" title="E26">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:10"/><i><font color="Green" title="E18">A24</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="uniroots.html#E15:10"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="uniroots.html#E16:10"><i><font color="Green" title="E17">A12</font></i></a>);<br/></i>
<a NAME="E18:10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">q</font>
 
;<br/>
<a NAME="E19:10"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c6">q</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c5">y2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="uniroots.html#E14:10"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="uniroots.html#E17:10"><i><font color="Green" title="E18">A24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:10"/>
 <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <font color="Maroon" title="c1">ps</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="uniroots.html#E3:10"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></i>;<br/>


</div>
