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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:15"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> or  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E3:15_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:15_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><a NAME="E6:15_1_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:15_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra1.html#T4">INTEGRA1:4</a></i>;<br/>
<b>thus </b><a NAME="E3:15_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:15_1_1_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E8:15_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:15_1_1_1_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_2"/>
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T33">FDIFF_1:33</a></i>;<br/><a NAME="E3:15_1_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E4:15_1_1_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/><a NAME="E5:15_1_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:15_1_1_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra1.html#T4">INTEGRA1:4</a></i>;<br/><a NAME="E8:15_1_1_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:15_1_1_1_2"/><i><font color="Green">E84</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E10:15_1_1_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/><a NAME="E11:15_1_1_1_2"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E12:15_1_1_1_2"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:15_1_1_1_2"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:15_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E16:15_1_1_1_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="rolle.html#T3">ROLLE:3</a></i>;<br/><a NAME="E17:15_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E18:15_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E19:15_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><b>hence </b><a NAME="E20:15_1_1_1_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font> holds <br/> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>and </b><br/><a NAME="E2:15_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:15_2"/><i><font color="Green">E92</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E4:15_2"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:15_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, , <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>
<i><font color="Green">E95</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E7:15_2"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<i><font color="Green">E97</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> = 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, </i>
<br/>.= 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E9:15_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:15_2"/>
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:15"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:15_3"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T28">PARTFUN1:28</a></i>;<br/>
<a NAME="E4:15_3"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:15_3"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <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"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:15_3"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <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"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E9:15_3"/><i><font color="Green">E102</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E10:15_3"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E11:15_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:15_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E105</font></i>: <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></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"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">A</font> <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"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E5:15_3_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:15_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:15_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>
<a NAME="E13:15_3"/><b>then </b><i><font color="Green">E106</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="ref" href="integra4.html#T18">INTEGRA4:18</a></i>;<br/>
<a NAME="E14:15_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:15_3_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E108</font></i>: <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></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"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon">A</font> <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"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span></span>)</span>

;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E5:15_3_2"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:15_3_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:15_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E16:15_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="integra4.html#T19">INTEGRA4:19</a></i>;<br/>
<b>hence </b><a NAME="E17:15_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:15"/><i><font color="Green">E109</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E9:15"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><b>proof </b></a><div class="add">

<a NAME="E1:15_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:15_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:15_4"/><i><font color="Green">E110</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E5:15_4"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <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>hence </b><a NAME="E6:15_4"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">k1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:15"/><i><font color="Green">E111</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</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">k1</font> holds <br/><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">s2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E14:15"/><i><font color="Green">E112</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</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">k1</font> holds <br/><font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E15:15"/><i><font color="Green">E113</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> <font color="Maroon">k1</font> holds <br/> <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Olive">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5"><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="E1:15_5"/><i><font color="Green">E114</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">k1</font>
 ;<br/>

<a NAME="E2:15_5"/>
<font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E3:15_5"/><b>then </b>
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:15_5"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font> <a href="tarski.html#R1">c=</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">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E5:15_5"/><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">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:15_5"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:15_5"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_5_1"/>
( 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">k1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:15_5_1"/><b>then </b>
( 1 <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E3:15_5_1"/><b>then </b>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <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">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:15_5_1"/>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:15_5"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_5_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_5_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_5_2_1_1"><b>suppose </b></a><a NAME="E1:15_5_2_1_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:15_5_2_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:15_5_2_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_5_2_1_2"><b>suppose </b></a><a NAME="E1:15_5_2_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:15_5_2_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:15_5_2_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:15_5_2_1_2"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E5:15_5_2_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:15_5_2_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:15_5"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E16:15"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6"><b>proof </b></a><div class="add">

<a NAME="E1:15_6"/>
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">k1</font> holds <br/><font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_6_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="E1:15_6_1"/><i><font color="Green">E133</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">k1</font>
 ;<br/>

<a NAME="E2:15_6_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:15_6_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:15_6_1"/>
<font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_6"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:15"/><i><font color="Green">E134</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</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#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7"><b>proof </b></a><div class="add">

<a NAME="E1:15_7"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E2:15_7"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:15_7"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E4:15_7"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:15_7"/><i><font color="Green">E137</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>hence </b><a NAME="E6:15_7"/><i><font color="Green">E138</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>

<b>thus </b><a NAME="E7:15_7"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/>

<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="E8:15_7"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 ;<br/>

<a NAME="E9:15_7"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_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"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:15_7_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:15_7"/><i><font color="Green">E141</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:15_7_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 or  not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_7_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 <b>by </b><i><a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:15_7"/><i><font color="Green">E142</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E12:15_7"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_7_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1_1"><b>suppose </b></a><a NAME="E1:15_7_3_1_1"/><i><font color="Green">E144</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:15_7_3_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:15_7_3_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:15_7_3_1_1"/><i><font color="Green">E147</font></i>: 
( <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">D</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:15_7_3_1_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E6:15_7_3_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E7:15_7_3_1_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E8:15_7_3_1_1"/>
<font color="Maroon">s2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E9:15_7_3_1_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><a NAME="E10:15_7_3_1_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E11:15_7_3_1_1"/>
<font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/><a NAME="E12:15_7_3_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E13:15_7_3_1_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E14:15_7_3_1_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1_2"><b>suppose </b></a><a NAME="E1:15_7_3_1_2"/><i><font color="Green">E151</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/><div class="add"><a NAME="E2:15_7_3_1_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_1"/>
1 <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K10">-</a> 1
 
;<br/>
<b>hence </b><a NAME="E2:15_7_3_1_2_1"/>
<font color="Maroon">k1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_7_3_1_2_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> or ( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:15_7_3_1_2_2_1_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:15_7_3_1_2_2_1_1"/><i><font color="Green">E154</font></i>: 
( <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">D</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_1_1"/><i><font color="Green">E155</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 2 <a href="tarski.html#R1">c=</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">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E2:15_7_3_1_2_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>hence </b><a NAME="E4:15_7_3_1_2_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:15_7_3_1_2_2_1_1"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_1_2"/><i><font color="Green">E157</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<b>thus </b><a NAME="E2:15_7_3_1_2_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 <b>by </b><i/>;<br/>

<b>hence </b><a NAME="E4:15_7_3_1_2_2_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:15_7_3_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:15_7_3_1_2_2_1_1"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:15_7_3_1_2_2_1_1"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_1_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E2:15_7_3_1_2_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:15_7_3_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:15_7_3_1_2_2_1_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:15_7_3_1_2_2_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><a NAME="E2:15_7_3_1_2_2_1_2"/><i><font color="Green">E161</font></i>: 
( <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">D</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:15_7_3_1_2_2_1_2_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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>

<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:15_7_3_1_2_2_1_2"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_7_3_1_2_2_1_2_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, </i>;<br/>

<a NAME="E2:15_7_3_1_2_2_1_2_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_2_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E4:15_7_3_1_2_2_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E5:15_7_3_1_2_2_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:15_7_3_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:15_7_3_1_2_2_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E6:15_7_3_1_2_2_1_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 &amp; <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_2_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:15_7_3_1_2_2_1_2_3"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>

<b>thus </b><a NAME="E2:15_7_3_1_2_2_1_2_3"/>
<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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/>

<a NAME="E3:15_7_3_1_2_2_1_2_3"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:15_7_3_1_2_2_1_2_3"/><b>then </b>
( <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> )
 
;<br/>
<b>hence </b><a NAME="E5:15_7_3_1_2_2_1_2_3"/>
<font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>

<b>hence </b><a NAME="E6:15_7_3_1_2_2_1_2_3"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E7:15_7_3_1_2_2_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:15_7_3_1_2_2_1_2"/><i><font color="Green">E166</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E2:15_7_3_1_2_2_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:15_7_3_1_2_2_1_2"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:15_7_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E10:15_7_3_1_2_2_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E11:15_7_3_1_2_2_1_2"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_7_3_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:15_7_3_1_2_2_1_3"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 ;<br/><div class="add"><a NAME="E2:15_7_3_1_2_2_1_3"/><i><font color="Green">E168</font></i>: 
( <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">D</font></span>)</span> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_3_1"/><i><font color="Green">E169</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K23">+</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><a NAME="E2:15_7_3_1_2_2_1_3_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"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>

<a NAME="E3:15_7_3_1_2_2_1_3_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:15_7_3_1_2_2_1_3_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:15_7_3_1_2_2_1_3"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_3_2"/><i><font color="Green">E171</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">D</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E2:15_7_3_1_2_2_1_3_2"/><b>then </b><i><font color="Green">E172</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#NR3" target="_self">&lt;</a> <font color="Maroon">k1</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:15_7_3_1_2_2_1_3_2"/><b>then </b><i><font color="Green">E173</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> <font color="Maroon">k1</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:15_7_3_1_2_2_1_3_2"/><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">s1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:15_7_3_1_2_2_1_3_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>thus </b><a NAME="E6:15_7_3_1_2_2_1_3_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>

<b>consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:15_7_3_1_2_2_1_3_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E9:15_7_3_1_2_2_1_3_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:15_7_3_1_2_2_1_3_2"/><b>then </b>
not <font color="Maroon">i</font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E11:15_7_3_1_2_2_1_3_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, <a class="ref" href="nat_2.html#T31">NAT_2:31</a></i>;<br/>
<a NAME="E12:15_7_3_1_2_2_1_3_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> <a href="binop_2.html#K10">-</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:15_7_3_1_2_2_1_3_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k1</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:15_7_3_1_2_2_1_3_2"/>
<font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="integra5.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>

<a NAME="E15:15_7_3_1_2_2_1_3_2"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <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">s2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>thus </b><a NAME="E16:15_7_3_1_2_2_1_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/>

<b>thus </b><a NAME="E17:15_7_3_1_2_2_1_3_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s2</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:15_7_3_1_2_2_1_3"/><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_3_3"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E2:15_7_3_1_2_2_1_3_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:15_7_3_1_2_2_1_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:15_7_3_1_2_2_1_3"/><i><font color="Green">E177</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_7_3_1_2_2_1_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:15_7_3_1_2_2_1_3_4"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <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">s2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:15_7_3_1_2_2_1_3_4"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:15_7_3_1_2_2_1_3_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:15_7_3_1_2_2_1_3_4"/><b>then </b>
( <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> 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> <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"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">s2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E5:15_7_3_1_2_2_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T36">FINSEQ_1:36</a></i>;<br/>
<a NAME="E6:15_7_3_1_2_2_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E7:15_7_3_1_2_2_1_3_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:15_7_3_1_2_2_1_3_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:15_7_3_1_2_2_1_3"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:15_7_3_1_2_2_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="binop_2.html#K10">-</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:15_7_3_1_2_2_1_3"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:15_7_3_1_2"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:15_7"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:15"/><i><font color="Green">E179</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_8"><b>proof </b></a><div class="add">

<a NAME="E1:15_8"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T24">INTEGRA1:24</a></i>;<br/>
<a NAME="E2:15_8"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">s2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/>
<a NAME="E3:15_8"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T106">RVSUM_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:15_8"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s2</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:15"/><i><font color="Green">E180</font></i>: 
 <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_9"><b>proof </b></a><div class="add">

<a NAME="E1:15_9"/><i><font color="Green">E181</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_9_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:15_9_1"/><i><font color="Green">E183</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:15_9_1"/><i><font color="Green">E184</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:15_9_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:15_9_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E7:15_9_1"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E8:15_9_1"/><i><font color="Green">E186</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<a NAME="E9:15_9_1"/>
<font color="Maroon">k</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">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:15_9_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E11:15_9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_9"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_9_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_9_2"/><i><font color="Green">E187</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">D</font></span>)</span> holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_9_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_9_2_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">k</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">D</font></span>)</span>
 ;<br/>

<a NAME="E2:15_9_2_1"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:15_9_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:15_9_2_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:15_9_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> =  <a href="integra1.html#K15">lower_volume</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E5:15_9_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<b>hence </b><a NAME="E6:15_9_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D10">INTEGRA1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_9"/>
 <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E20:15"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_10"><b>proof </b></a><div class="add">

<a NAME="E1:15_10"/><i><font color="Green">E191</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_10_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:15_10_1"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:15_10_1"/><i><font color="Green">E194</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:15_10_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:15_10_1"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> is <a href="seq_4.html#V2">bounded_below</a> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<a NAME="E7:15_10_1"/><b>then </b><i><font color="Green">E195</font></i>: 
 <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E8:15_10_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<a NAME="E9:15_10_1"/><b>then </b><i><font color="Green">E196</font></i>: 
<span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E10:15_10_1"/>
<font color="Maroon">k</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">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:15_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:15_10"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_10_2"><b>proof </b></a><div class="add">

<a NAME="E1:15_10_2"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</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">D</font></span>)</span> holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_10_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:15_10_2_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">k</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">D</font></span>)</span>
 ;<br/>

<a NAME="E2:15_10_2_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:15_10_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:15_10_2_1"/>
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:15_10_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> =  <a href="integra1.html#K14">upper_volume</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E5:15_10_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<b>hence </b><a NAME="E6:15_10_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D9">INTEGRA1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_10"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:15"/>
(  <a href="integra1.html#K7">lower_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K6">upper_sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="fdiff_1.html#K2">`|</a> <font color="Maroon">X</font></span>)</span> <a href="integra5.html#K1" target="_self">||</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>


</div>
