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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">T</font> be   <a href="integra2.html#NM1" title="INTEGRA2:NM.1">DivSequence</a> of <font color="Maroon" title="c1">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E33">A2</font></i>: 
 <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font> is <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E4:32"/><i><font color="Green" title="E35">A4</font></i>: 
(  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a>  &amp;  <a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green" title="E36">A5</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green" title="E37">A6</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E4:32_1"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ;<br/>

<a NAME="E2:32_1_1"/><b>then </b><i><font color="Green" title="E38">A7</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<a NAME="E3:32_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11" title="ABSVALUE:th.11">ABSVALUE:11</a></i>;<br/>
<a NAME="E4:32_1_1"/><b>then </b><i><font color="Green" title="E39">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_1_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:32_1_1"/><i><font color="Green" title="E40">A9</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/>
<a NAME="E6:32_1_1"/><i><font color="Green" title="E41">A10</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra2.html#D3" title="INTEGRA2:def.3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E7:32_1_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></i>;<br/>
<a NAME="E8:32_1_1"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:32_1_1"/><i><font color="Green" title="E42">A11</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> &amp;  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E11:32_1_1"/><i><font color="Green" title="E43">A12</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:32_1_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">D</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E13:32_1_1"/><i><font color="Green" title="E44">A13</font></i>: 
<font color="Maroon" title="c8">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font>
 ;<br/>
<a NAME="E14:32_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_1_1"><i><font color="Green" title="E43">A12</font></i></a>, <a class="txt" href="integra3.html#E13:32_1_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E15:32_1_1"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font></span>)</span>,<font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:32_1_1"><i><font color="Green" title="E42">A11</font></i></a>, <a class="txt" href="integra3.html#E13:32_1_1"><i><font color="Green" title="E44">A13</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/>
<b>hence </b><a NAME="E16:32_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_1_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="integra3.html#E5:32_1_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="integra3.html#E6:32_1_1"><i><font color="Green" title="E41">A10</font></i></a>, <a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:32"/><i><font color="Green" title="E37">A14</font></i>: 
for <font color="Olive" title="b1">D</font>, <font color="Olive" title="b2">D1</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b3">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp; <font color="Olive" title="b2">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">D</font>, <font color="Maroon" title="c5">D1</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">D2</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E2:32_2"/><i><font color="Green" title="E38">A15</font></i>: 
( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c6">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self" title="INTEGRA3:th.3">Th3</a></i>;<br/>
<a NAME="E3:32_2"/><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E4:32_2"/><b>then </b>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="integra1.html#T48" title="INTEGRA1:th.48">INTEGRA1:48</a></i>;<br/>
<a NAME="E5:32_2"/><b>then </b><i><font color="Green" title="E40">A17</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E6:32_2"/>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="integra3.html#E3:32_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="integra1.html#T48" title="INTEGRA1:th.48">INTEGRA1:48</a></i>;<br/>
<a NAME="E7:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2"/>
 ex <font color="Olive" title="b1">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="integra3.html#E5:32_2"><i><font color="Green" title="E40">A17</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32"/><i><font color="Green" title="E38">A18</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E12"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green" title="E39">A19</font></i>: 
for <font color="Olive" title="b1">D</font>, <font color="Olive" title="b2">D1</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>  st  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Olive" title="b2">D1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Olive" title="b1">D</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b3">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp; <font color="Olive" title="b2">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b3">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b2">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b3">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Olive" title="b2">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">D</font>, <font color="Maroon" title="c5">D1</font> be    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E1:32_3"/><i><font color="Green" title="E40">A20</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3"/>
 ex <font color="Olive" title="b1">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">D2</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E2:32_3_1"/><i><font color="Green" title="E41">A21</font></i>: 
( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c6">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/>
<a NAME="E3:32_3_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>,  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="integra1.html#K18" title="INTEGRA1:func.18">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,$1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $2;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>) <b>-&gt; </b>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  =  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,$1;<br/>
<a NAME="E1:32_3_1_1"/><i><font color="Green" title="E42">A22</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1"/><i><font color="Green" title="E43">A23</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1"/><i><font color="Green" title="E44">A24</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b2">j</font> holds <br/><font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:32_3_1_1_1_1"/><i><font color="Green" title="E45">A25</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<b>assume </b><a NAME="E2:32_3_1_1_1_1"/><i><font color="Green" title="E46">A26</font></i>: 
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> )
 ;<br/>

<b>assume </b><a NAME="E3:32_3_1_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 2
 ;<br/>

<a NAME="E4:32_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E5:32_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E47">A27</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_1"/><i><font color="Green" title="E48">A28</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_1"><i><font color="Green" title="E47">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_1"/><b>then </b><i><font color="Green" title="E49">A29</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_1_1_1_1"/><i><font color="Green" title="E50">A30</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E51">A31</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E52">A32</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 )
 <b>by </b><i><a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E50">A30</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E53">A33</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_1_1_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="integra3.html#E14"><i><font color="Green" title="E14">Lm6</font></i></a></i>;<br/><a NAME="E12:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E54">A34</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E53">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A31</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E55">A35</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_1_1_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E51">A31</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E55">A35</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_1_1_1_1_1_1"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_1_1_1_1"><i><font color="Green" title="E54">A34</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_1_1_1_2"/><i><font color="Green" title="E50">A36</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E51">A37</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A38</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E50">A36</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E52">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E53">A39</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_1_1_2"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="integra3.html#E14"><i><font color="Green" title="E14">Lm6</font></i></a></i>;<br/><a NAME="E11:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E54">A40</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E53">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_1_1_2"/><i><font color="Green" title="E55">A41</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E56">A42</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_1_1_2"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E55">A41</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E56">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_1_1_1_1_1_2"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c4">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_1_1_1_2"><i><font color="Green" title="E54">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E12:32_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green" title="E40">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> implies  ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">$1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> ) );<br/>
<a NAME="E3:32_3_1_1_1"/><i><font color="Green" title="E45">A43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E46">A44</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:32_3_1_1_1_2"/><i><font color="Green" title="E47">A45</font></i>: 
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T2" target="_self" title="INTEGRA3:th.2">Th2</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1"/><i><font color="Green" title="E48">A46</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2"><i><font color="Green" title="E46">A44</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E49">A47</font></i>: 
( <font color="Maroon" title="c8">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c8">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">j1</font> = <font color="Maroon" title="c8">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E50">A48</font></i>: 
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1"/><i><font color="Green" title="E51">A49</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_2_1_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_2_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1_1_2_1"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T37" title="INTEGRA1:th.37">INTEGRA1:37</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E52">A50</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1"/><i><font color="Green" title="E53">A51</font></i>: 
( <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_2_1_2"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_2_1_2"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:32_3_1_1_1_2_1"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c5">D1</font>,1,<font color="Maroon" title="c9">j1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T37" title="INTEGRA1:th.37">INTEGRA1:37</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E54">A52</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1"/><i><font color="Green" title="E55">A53</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_3"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_1"/><i><font color="Green" title="E56">A54</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green" title="E57">A55</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green" title="E58">A56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green" title="E59">A57</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E57">A55</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_3_1"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green" title="E60">A58</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_3_1"/><i><font color="Green" title="E61">A59</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_3_1"/><i><font color="Green" title="E62">A60</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_1_1"/><i><font color="Green" title="E63">A61</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1_1"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_2"><i><font color="Green" title="E46">A44</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1_1"><i><font color="Green" title="E63">A61</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1"><i><font color="Green" title="E50">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_2_1_3_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E60">A58</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1_3_1"/><i><font color="Green" title="E63">A62</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> implies <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1_2"/><i><font color="Green" title="E64">A63</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E61">A59</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1_2"/><b>then </b><i><font color="Green" title="E65">A64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_2_1_3_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E62">A60</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_3_1_2"><i><font color="Green" title="E64">A63</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_3_1_2"><i><font color="Green" title="E65">A64</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_3_1_1_1_2_1_3_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> implies <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_1_3"/><i><font color="Green" title="E64">A65</font></i>: 
( <font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">m</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1_3"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1_3"/><b>then </b><i><font color="Green" title="E65">A66</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E60">A58</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1_3"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_1_3"/><b>then </b><i><font color="Green" title="E66">A67</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E61">A59</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_3_1_3"><i><font color="Green" title="E65">A66</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_3_1_3"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">m</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1_3"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_3_1_3"><i><font color="Green" title="E66">A67</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:32_3_1_1_1_2_1_3_1"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E56">A54</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E57">A55</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E59">A57</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1_3_1"><i><font color="Green" title="E63">A62</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_3"/><b>then </b><i><font color="Green" title="E56">A68</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_3"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_2"/><i><font color="Green" title="E57">A69</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green" title="E58">A70</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green" title="E59">A71</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:32_3_1_1_1_2_1_3_2"/><i><font color="Green" title="E60">A72</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c10">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_3_2"/><i><font color="Green" title="E61">A73</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E57">A69</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E59">A71</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E58">A70</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E59">A71</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green" title="E62">A74</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E60">A72</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E61">A73</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1_3_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E60">A72</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green" title="E63">A75</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E62">A74</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green" title="E64">A76</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E63">A75</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E22:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon" title="c10">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E60">A72</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_2_1_3_2"><i><font color="Green" title="E64">A76</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_2_1_3"/><b>then </b>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_2_1_3"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_3"><i><font color="Green" title="E56">A68</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E56">A77</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1"><i><font color="Green" title="E54">A52</font></i></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1"/><i><font color="Green" title="E57">A78</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> holds <br/><font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_4"/><i><font color="Green" title="E58">A79</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> )
 ;<br/>

<b>assume </b><a NAME="E2:32_3_1_1_1_2_1_4"/><i><font color="Green" title="E59">A80</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_4_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_4"><i><font color="Green" title="E59">A80</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_4_1_1_1"/><i><font color="Green" title="E60">A81</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_4_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E61">A82</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E62">A83</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E63">A84</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E61">A82</font></i></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E64">A85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_4_1_1_1"/><i><font color="Green" title="E65">A86</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E60">A81</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E63">A84</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E66">A87</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E60">A81</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E61">A82</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E13:32_3_1_1_1_2_1_4_1_1_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E63">A84</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E65">A86</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E14:32_3_1_1_1_2_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1"><i><font color="Green" title="E56">A77</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E62">A83</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E64">A85</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_4_1_1_1"><i><font color="Green" title="E66">A87</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_4_1_1_2"/><i><font color="Green" title="E60">A88</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E61">A89</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_4_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E62">A90</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E63">A91</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/><a NAME="E7:32_3_1_1_1_2_1_4_1_1_2"/>
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E62">A90</font></i></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E64">A92</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E60">A88</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E65">A93</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E60">A88</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E63">A91</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E12:32_3_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_4"><i><font color="Green" title="E58">A79</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E64">A92</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E13:32_3_1_1_1_2_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1"><i><font color="Green" title="E56">A77</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E61">A89</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E63">A91</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_4_1_1_2"><i><font color="Green" title="E65">A93</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:32_3_1_1_1_2_1"/><i><font color="Green" title="E58">A94</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1"><i><font color="Green" title="E52">A50</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1"><i><font color="Green" title="E54">A52</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1"><i><font color="Green" title="E55">A53</font></i></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E59">A95</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_2_1"><i><font color="Green" title="E58">A94</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1"/>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E60">A96</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E17:32_3_1_1_1_2_1"><i><font color="Green" title="E59">A95</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_2_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E22:32_3_1_1_1_2_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E23:32_3_1_1_1_2_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E24:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E61">A97</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_2_1"><i><font color="Green" title="E60">A96</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E25:32_3_1_1_1_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_5"/><i><font color="Green" title="E62">A98</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">k</font> = <font color="Maroon" title="c10">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_5"/><i><font color="Green" title="E63">A99</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green" title="E64">A100</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_5"><i><font color="Green" title="E62">A98</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green" title="E65">A101</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_5"><i><font color="Green" title="E64">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green" title="E66">A102</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_5"/><i><font color="Green" title="E67">A103</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_5_1"/><i><font color="Green" title="E68">A104</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_5_1"/><i><font color="Green" title="E69">A105</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_5"><i><font color="Green" title="E66">A102</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_5_1"/><b>then </b><i><font color="Green" title="E70">A106</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_5_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_5_1_1_1"/>
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_5_1_1_1_1"/><i><font color="Green" title="E71">A107</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E72">A108</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E69">A105</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_5_1_1_1_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_5_1_1_1_1"><i><font color="Green" title="E71">A107</font></i></a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_2_1_5_1_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E70">A106</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1_1_1_1"><i><font color="Green" title="E72">A108</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_5_1_1_1_2"/><i><font color="Green" title="E71">A109</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E72">A110</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E69">A105</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_5_1_1_1_2"/><i><font color="Green" title="E73">A111</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E69">A105</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E71">A109</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">k1</font> = <font color="Maroon" title="c11">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E69">A105</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E71">A109</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E7:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E74">A112</font></i>: 
<font color="Maroon" title="c12">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_5"><i><font color="Green" title="E64">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E73">A111</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E75">A113</font></i>: 
<font color="Maroon" title="c12">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c12">k1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E74">A112</font></i></a></i>;<br/><a NAME="E11:32_3_1_1_1_2_1_5_1_1_1_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_5"><i><font color="Green" title="E64">A100</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E71">A109</font></i></a></i>;<br/><a NAME="E12:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E76">A114</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E70">A106</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E13:32_3_1_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c12">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_5"><i><font color="Green" title="E64">A100</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E75">A113</font></i></a></i>;<br/><b>hence </b><a NAME="E14:32_3_1_1_1_2_1_5_1_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E69">A105</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E72">A110</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E73">A111</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_5_1_1_1_2"><i><font color="Green" title="E76">A114</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_3_1_1_1_2_1_5_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_5_1"><i><font color="Green" title="E68">A104</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_2_1_5"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_5"><i><font color="Green" title="E63">A99</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_5"/><b>then </b>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E68">A115</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_5"><i><font color="Green" title="E65">A101</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_5"><i><font color="Green" title="E66">A102</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_5"><i><font color="Green" title="E67">A103</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E12:32_3_1_1_1_2_1_5"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_5"><i><font color="Green" title="E64">A100</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_5"><i><font color="Green" title="E65">A101</font></i></a></i>;<br/>
<a NAME="E13:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green" title="E69">A116</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_2_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green" title="E70">A117</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1_5"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1_5"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green" title="E71">A118</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_2_1"><i><font color="Green" title="E57">A78</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_5"><i><font color="Green" title="E64">A100</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1_5"><i><font color="Green" title="E69">A116</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_2_1_5"><i><font color="Green" title="E71">A118</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c11">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E16:32_3_1_1_1_2_1_5"><i><font color="Green" title="E70">A117</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E21:32_3_1_1_1_2_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_5"><i><font color="Green" title="E68">A115</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E62">A119</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_2_1"><i><font color="Green" title="E61">A97</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E27:32_3_1_1_1_2_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E63">A120</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E29:32_3_1_1_1_2_1"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E64">A121</font></i>: 
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<i><font color="Green" title="E65">A122</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E28:32_3_1_1_1_2_1"><i><font color="Green" title="E63">A120</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>)
<b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_2_1"><i><font color="Green" title="E62">A119</font></i></a>, <a class="txt" href="integra3.html#E30:32_3_1_1_1_2_1"><i><font color="Green" title="E64">A121</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>
;<br/>
<a NAME="E32:32_3_1_1_1_2_1"/><i><font color="Green" title="E66">A123</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_7"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_7"/><b>then </b>
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_7"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="integra3.html#T7" target="_self" title="INTEGRA3:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_7"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:32_3_1_1_1_2_1"/><i><font color="Green" title="E67">A124</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8"/><i><font color="Green" title="E68">A125</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green" title="E69">A126</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8"/><i><font color="Green" title="E70">A127</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E71">A128</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ID2</font> = <font color="Maroon" title="c10">ID1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E72">A129</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">ID1</font> &amp; <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">ID2</font> &amp; <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_2_1_8_2_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">ID1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<b>thus </b><a NAME="E2:32_3_1_1_1_2_1_8_2_1"/>
<font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">ID2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<a NAME="E3:32_3_1_1_1_2_1_8_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E71">A128</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_1"/>
<font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E73">A130</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_2"/><b>then </b><i><font color="Green" title="E74">A131</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E75">A132</font></i>: 
<font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">ID1</font> &amp; <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E72">A129</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">ID1</font> &amp; <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E74">A131</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_2_2"/><b>then </b>
<font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_2"/>
<font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E76">A133</font></i>: 
<font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_3"/>
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ID2</font> &amp; <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E72">A129</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E74">A131</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ID2</font> &amp; <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_2_3"/><b>then </b>
<font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_3"/>
<font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_2_1_8_2"/><b>then </b><i><font color="Green" title="E77">A134</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E72">A129</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E73">A130</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E75">A132</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E78">A135</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_4"/><i><font color="Green" title="E79">A136</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E75">A132</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E76">A133</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_4"/><i><font color="Green" title="E80">A137</font></i>: 
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_2_4"/>
( not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> &amp; not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1"/><i><font color="Green" title="E81">A138</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_2_4_1"><i><font color="Green" title="E81">A138</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_8_2_4_1_1_1"/><i><font color="Green" title="E82">A139</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_8_2_4_1_1_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E77">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_2_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_2_4"><i><font color="Green" title="E80">A137</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_2_4_1_1_1"><i><font color="Green" title="E82">A139</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_8_2_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E7:32_3_1_1_1_2_1_8_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_8_2_4_1_1_2"/><i><font color="Green" title="E82">A140</font></i>: 
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_8_2_4_1_1_2"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E77">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_2_4_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_2_4"><i><font color="Green" title="E80">A137</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_2_4_1_1_2"><i><font color="Green" title="E82">A140</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_8_2_4_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E7:32_3_1_1_1_2_1_8_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_4"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_2_4"><i><font color="Green" title="E79">A136</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E79">A141</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_5"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_2_5"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:32_3_1_1_1_2_1_8_2"/><i><font color="Green" title="E80">A142</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_6"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E77">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_6"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E79">A141</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_2_6"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E78">A135</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1_8_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_7"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E77">A134</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_7"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E79">A141</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_2_7"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E78">A135</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_3_1_1_1_2_1_8_2"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">ID1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">ID2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E80">A142</font></i></a>, <a class="ref" href="integra3.html#T4" target="_self" title="INTEGRA3:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E15:32_3_1_1_1_2_1_8_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E72">A129</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E75">A132</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_8_2"><i><font color="Green" title="E76">A133</font></i></a>, <a class="ref" href="goboard2.html#T19" title="GOBOARD2:th.19">GOBOARD2:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_2_1_8"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_3"/><i><font color="Green" title="E71">A143</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<b>thus </b><a NAME="E4:32_3_1_1_1_2_1_8_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8"><i><font color="Green" title="E70">A127</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_3"><i><font color="Green" title="E71">A143</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green" title="E71">A144</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 )
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8"/><i><font color="Green" title="E72">A145</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1_1_2_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green" title="E73">A146</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8"><i><font color="Green" title="E72">A145</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_8"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8"><i><font color="Green" title="E72">A145</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_8"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_8"><i><font color="Green" title="E73">A146</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green" title="E74">A147</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_8"/><i><font color="Green" title="E75">A148</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1_8"/><i><font color="Green" title="E76">A149</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_8_6_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_8"><i><font color="Green" title="E71">A144</font></i></a>, <a class="txt" href="integra3.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green" title="E77">A150</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green" title="E78">A151</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E77">A150</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green" title="E79">A152</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_1_2_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_6_1_1_2_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_1_2"/><b>then </b><i><font color="Green" title="E80">A153</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_1_2"/><i><font color="Green" title="E81">A154</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E78">A151</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_1_2"/><i><font color="Green" title="E82">A155</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_8_6_1_1_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E78">A151</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_6_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_6_1_1_2"><i><font color="Green" title="E81">A154</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_6_1_1_2"><i><font color="Green" title="E82">A155</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_2_1_8_6_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_1_2"><i><font color="Green" title="E80">A153</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:32_3_1_1_1_2_1_8_6_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_6_1_1"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E77">A150</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then </b><i><font color="Green" title="E80">A156</font></i>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_8"><i><font color="Green" title="E75">A148</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_8"><i><font color="Green" title="E74">A147</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1_8"><i><font color="Green" title="E76">A149</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E78">A151</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E79">A152</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
;<br/><a NAME="E7:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green" title="E81">A157</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra3.html#T8" target="_self" title="INTEGRA3:th.8">Th8</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_8_6_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_8_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E81">A157</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_1_1_2_1_8_6_1_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8_6_1_1"><i><font color="Green" title="E80">A156</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E77">A158</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E78">A159</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_1"/><i><font color="Green" title="E79">A160</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E77">A158</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2_1"><i><font color="Green" title="E79">A160</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E77">A158</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E78">A159</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b><i><font color="Green" title="E79">A161</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T107" title="RVSUM_1:th.107">RVSUM_1:107</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E80">A162</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2"/><i><font color="Green" title="E81">A163</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2"/><i><font color="Green" title="E82">A164</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E77">A158</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_2_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_2"/><i><font color="Green" title="E83">A165</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green" title="E84">A166</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_2_2"><i><font color="Green" title="E82">A164</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_2_2_3"/><b>then </b><i><font color="Green" title="E85">A167</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green" title="E86">A168</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green" title="E87">A169</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green" title="E88">A170</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E8:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green" title="E89">A171</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>thus </b><a NAME="E9:32_3_1_1_1_2_1_8_6_1_2_2_3"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2_2_3"><i><font color="Green" title="E84">A166</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8_6_1_2_2_3"><i><font color="Green" title="E87">A169</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_8_6_1_2_2_3"><i><font color="Green" title="E88">A170</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E10:32_3_1_1_1_2_1_8_6_1_2_2_3"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_6_1_2_2_3"><i><font color="Green" title="E85">A167</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_8_6_1_2_2_3"><i><font color="Green" title="E86">A168</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_8_6_1_2_2_3"><i><font color="Green" title="E89">A171</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_2_2"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2_2"><i><font color="Green" title="E81">A163</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_2_2"><i><font color="Green" title="E82">A164</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<b>then </b> <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_6_1_2_2"><i><font color="Green" title="E83">A165</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_6_1_2_2"><i><font color="Green" title="E83">A165</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_2_1_8_6_1_2_2"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b><i><font color="Green" title="E81">A172</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1_8"><i><font color="Green" title="E76">A149</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E82">A173</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E83">A174</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E84">A175</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_3"/><i><font color="Green" title="E85">A176</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_3"/><i><font color="Green" title="E86">A177</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8"><i><font color="Green" title="E68">A125</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8"><i><font color="Green" title="E69">A126</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E77">A158</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_3_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green" title="E87">A178</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E85">A176</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c11">ID2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2;<br/>
<i><font color="Green" title="E88">A179</font></i>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E79">A161</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E87">A178</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_8"><i><font color="Green" title="E75">A148</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E5:32_3_1_1_1_2_1_8_6_1_2_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E85">A176</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E86">A177</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green" title="E89">A180</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_2_1_8"><i><font color="Green" title="E75">A148</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green" title="E90">A181</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E82">A173</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_8_6_1_2_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/>
<a NAME="E13:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green" title="E91">A182</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E82">A173</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E83">A174</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E83">A174</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E88">A179</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E89">A180</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E90">A181</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green" title="E92">A183</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">A</font> =  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<b>set </b><font color="Maroon" title="c13">B</font> =  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">C</font> =  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E20:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">A</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">A</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E81">A172</font></i></a></i>;<br/>
<a NAME="E21:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_8"><i><font color="Green" title="E74">A147</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E91">A182</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E22:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E23:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E91">A182</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E25:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_2_1_8_6_1_2_3"><i><font color="Green" title="E92">A183</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E26:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E11:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green" title="E85">A184</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="integra3.html#E14"><i><font color="Green" title="E14">Lm6</font></i></a></i>;<br/><a NAME="E12:32_3_1_1_1_2_1_8_6_1_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E13:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E85">A184</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E14:32_3_1_1_1_2_1_8_6_1_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E80">A162</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_2_1_8_6_1_2"><i><font color="Green" title="E84">A175</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:32_3_1_1_1_2_1_8"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E34:32_3_1_1_1_2_1"/><i><font color="Green" title="E68">A185</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_9"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_9"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_9"/><b>then </b><i><font color="Green" title="E69">A186</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_9"/><b>then </b><i><font color="Green" title="E70">A187</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_9"/><b>then </b><i><font color="Green" title="E71">A188</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) )
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_9"/><i><font color="Green" title="E72">A189</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_9"><i><font color="Green" title="E69">A186</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_9"/><i><font color="Green" title="E73">A190</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E32:32_3_1_1_1_2_1"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_9"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_9"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_9"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_9"><i><font color="Green" title="E71">A188</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_9"><i><font color="Green" title="E73">A190</font></i></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_9"><i><font color="Green" title="E70">A187</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E11:32_3_1_1_1_2_1_9"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_2_1_9"><i><font color="Green" title="E72">A189</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:32_3_1_1_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_10"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_10"/><i><font color="Green" title="E69">A191</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_10"/><b>then </b><i><font color="Green" title="E70">A192</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_10"/><b>then </b><i><font color="Green" title="E71">A193</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) )
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_10"/><i><font color="Green" title="E72">A194</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_2_1_10"><i><font color="Green" title="E69">A191</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_10"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_10"/><b>then </b><i><font color="Green" title="E73">A195</font></i>: 
<font color="Maroon" title="c9">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_10"/>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_10"/><b>then </b>
<font color="Maroon" title="c9">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_10"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c9">j1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_2_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_2_1_10"><i><font color="Green" title="E71">A193</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_2_1_10"><i><font color="Green" title="E73">A195</font></i></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_2_1_10"><i><font color="Green" title="E70">A192</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E11:32_3_1_1_1_2_1_10"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span>,<font color="Maroon" title="c8">j</font>,<font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>)
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_2_1_10"><i><font color="Green" title="E72">A194</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E36:32_3_1_1_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c8">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E31:32_3_1_1_1_2_1"><i><font color="Green" title="E65">A122</font></i></a>, <a class="txt" href="integra3.html#E33:32_3_1_1_1_2_1"><i><font color="Green" title="E67">A124</font></i></a>, <a class="txt" href="integra3.html#E34:32_3_1_1_1_2_1"><i><font color="Green" title="E68">A185</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:32_3_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_2"><i><font color="Green" title="E47">A45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1"/><i><font color="Green" title="E46">A196</font></i>: 
for <font color="Olive" title="b1">i</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3"/><i><font color="Green" title="E47">A197</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>]
 ;<br/>

<a NAME="E2:32_3_1_1_1_3"/><i><font color="Green" title="E48">A198</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1"/><i><font color="Green" title="E49">A199</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:32_3_1_1_1_3_1"/><i><font color="Green" title="E50">A200</font></i>: 
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T2" target="_self" title="INTEGRA3:th.2">Th2</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1"/><i><font color="Green" title="E51">A201</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1"><i><font color="Green" title="E51">A201</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E52">A202</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3"><i><font color="Green" title="E48">A198</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E53">A203</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:32_3_1_1_1_3_1"/><i><font color="Green" title="E54">A204</font></i>: 
( <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3"><i><font color="Green" title="E47">A197</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1"><i><font color="Green" title="E52">A202</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_3_1"/><i><font color="Green" title="E55">A205</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E13:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E56">A206</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_3_1"/><i><font color="Green" title="E57">A207</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_1"/><i><font color="Green" title="E58">A208</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">j</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_1_1_1"/>
( <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">j</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_1"><i><font color="Green" title="E58">A208</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1"><i><font color="Green" title="E53">A203</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E59">A209</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E60">A210</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E59">A209</font></i></a>, <a class="ref" href="integra3.html#T4" target="_self" title="INTEGRA3:th.4">Th4</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1"><i><font color="Green" title="E53">A203</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_1_1_1_1"><i><font color="Green" title="E60">A210</font></i></a>, <a class="ref" href="goboard2.html#T19" title="GOBOARD2:th.19">GOBOARD2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A211</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E60">A212</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_1_1_1_2"/><i><font color="Green" title="E61">A213</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_1_1_1_2"/><i><font color="Green" title="E62">A214</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E63">A215</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E59">A211</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E64">A216</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E60">A212</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_1_1_1_2"/><i><font color="Green" title="E65">A217</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_1_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_1_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E12:32_3_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E61">A213</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E63">A215</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E64">A216</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">A218</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E62">A214</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E65">A217</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E15:32_3_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E16:32_3_1_1_1_3_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1"><i><font color="Green" title="E53">A203</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1_1_1_1_2"><i><font color="Green" title="E66">A218</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E58">A219</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E59">A220</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_3_1"/><i><font color="Green" title="E60">A221</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_3_1"/><i><font color="Green" title="E61">A222</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E62">A223</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E63">A224</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E64">A225</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E22:32_3_1_1_1_3_1"/><i><font color="Green" title="E65">A226</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E23:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E24:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E66">A227</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E25:32_3_1_1_1_3_1"/><i><font color="Green" title="E67">A228</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:32_3_1_1_1_3_1"/><i><font color="Green" title="E68">A229</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:32_3_1_1_1_3_1"/>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E28:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E29:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E69">A230</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E30:32_3_1_1_1_3_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E31:32_3_1_1_1_3_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E70">A231</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1"><i><font color="Green" title="E67">A228</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>
;<br/>
<a NAME="E33:32_3_1_1_1_3_1"/><i><font color="Green" title="E71">A232</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1"/>
( <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> or <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E72">A233</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E73">A234</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) )
 <b>by </b><i><a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E74">A235</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green" title="E75">A236</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E73">A234</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
1

;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E73">A234</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E74">A235</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E75">A236</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E76">A237</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E77">A238</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E78">A239</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E76">A237</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E77">A238</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E79">A240</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_3"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_3"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E80">A241</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E81">A242</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E82">A243</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E19:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E83">A244</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><b>then </b><i><font color="Green" title="E84">A245</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E80">A241</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E81">A242</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E82">A243</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E80">A241</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span>

;<br/><a NAME="E21:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E85">A246</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5"/><b>then </b><i><font color="Green" title="E86">A247</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E87">A248</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E88">A249</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E89">A250</font></i>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E86">A247</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E88">A249</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E90">A251</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_2"/><i><font color="Green" title="E91">A252</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E89">A250</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_2"/>
not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/><i><font color="Green" title="E92">A253</font></i>: 
( <font color="Maroon" title="c11">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E87">A248</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E89">A250</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/><b>then </b>
( <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">k1</font> &amp; <font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_5_2_1"><i><font color="Green" title="E92">A253</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_5_2"><i><font color="Green" title="E91">A252</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E91">A254</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_3"/><i><font color="Green" title="E92">A255</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_3"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E87">A248</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E89">A250</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_3"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_5_3"><i><font color="Green" title="E92">A255</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_3"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E90">A251</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E92">A256</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_4"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E88">A249</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E86">A247</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green" title="E93">A257</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_5"/><i><font color="Green" title="E94">A258</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E92">A256</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_5"/>
not <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">k1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/><i><font color="Green" title="E95">A259</font></i>: 
( <font color="Maroon" title="c11">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E86">A247</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E92">A256</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/><b>then </b>
( <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">k1</font> &amp; <font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_5_5_1"><i><font color="Green" title="E95">A259</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_5"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_5_5"><i><font color="Green" title="E94">A258</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_1_5"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_6"/><i><font color="Green" title="E94">A260</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_6"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E86">A247</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E92">A256</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_6"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_5_6"><i><font color="Green" title="E94">A260</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_6"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E93">A257</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E91">A254</font></i></a>, <a class="ref" href="integra3.html#T4" target="_self" title="INTEGRA3:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E12:32_3_1_1_1_3_1_3_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E88">A249</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E89">A250</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_1_5"><i><font color="Green" title="E92">A256</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green" title="E86">A261</font></i>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_6"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E79">A240</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_6"/><b>then </b><i><font color="Green" title="E87">A262</font></i>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> or <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> implies <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_6_1"/><b>then </b><i><font color="Green" title="E88">A263</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E85">A246</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_6_1"><i><font color="Green" title="E88">A263</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_6"><i><font color="Green" title="E87">A262</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E86">A261</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><i><font color="Green" title="E87">A264</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><b>then </b>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 ;<br/><b>then </b><i><font color="Green" title="E88">A265</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E82">A243</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E83">A244</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E87">A264</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E89">A266</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><i><font color="Green" title="E90">A267</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/><i><font color="Green" title="E91">A268</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E87">A264</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/><b>then </b><i><font color="Green" title="E92">A269</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E87">A264</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"><i><font color="Green" title="E91">A268</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"><i><font color="Green" title="E92">A269</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><i><font color="Green" title="E91">A270</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E87">A264</font></i></a>, <a class="ref" href="integra3.html#T17" target="_self" title="INTEGRA3:th.17">Th17</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><b>then </b><i><font color="Green" title="E92">A271</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E89">A266</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E90">A267</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E91">A270</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E84">A245</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E87">A264</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E88">A265</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_1_7_1_1"><i><font color="Green" title="E92">A271</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E87">A272</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E88">A273</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E89">A274</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E90">A275</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E91">A276</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E92">A277</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E91">A276</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E93">A278</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><b>then </b><i><font color="Green" title="E94">A279</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E89">A274</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E91">A276</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 2 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E90">A275</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E91">A276</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E93">A278</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>

;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E84">A245</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E94">A279</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E95">A280</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T107" title="RVSUM_1:th.107">RVSUM_1:107</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green" title="E96">A281</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E92">A277</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/><b>then </b><i><font color="Green" title="E97">A282</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/><i><font color="Green" title="E98">A283</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E92">A277</font></i></a>, <a class="ref" href="integra3.html#T17" target="_self" title="INTEGRA3:th.17">Th17</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"><i><font color="Green" title="E97">A282</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"><i><font color="Green" title="E98">A283</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/><b>then </b><i><font color="Green" title="E97">A284</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/><i><font color="Green" title="E98">A285</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a>, <a class="ref" href="integra3.html#T17" target="_self" title="INTEGRA3:th.17">Th17</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"><i><font color="Green" title="E97">A284</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"><i><font color="Green" title="E98">A285</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b><i><font color="Green" title="E97">A286</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E95">A280</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E96">A281</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b><i><font color="Green" title="E98">A287</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E92">A277</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b><i><font color="Green" title="E99">A288</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E87">A272</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E88">A273</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><i><font color="Green" title="E100">A289</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"><i><font color="Green" title="E98">A287</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><i><font color="Green" title="E101">A290</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"><i><font color="Green" title="E99">A288</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E10:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E72">A233</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"><i><font color="Green" title="E100">A289</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"><i><font color="Green" title="E101">A290</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E17:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_1_7_1_2"><i><font color="Green" title="E97">A286</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E87">A291</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E78">A239</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E25:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E12"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/><a NAME="E26:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E88">A292</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E27:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_8"/><i><font color="Green" title="E89">A293</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_8"/><b>then </b><i><font color="Green" title="E90">A294</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_8"/>
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_1_8"><i><font color="Green" title="E89">A293</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_8"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_8"/><b>then </b><i><font color="Green" title="E91">A295</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_1_8"><i><font color="Green" title="E90">A294</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_8"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_8"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_1_8"><i><font color="Green" title="E91">A295</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E88">A292</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E29:32_3_1_1_1_3_1_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1_3_1_1_1"><i><font color="Green" title="E87">A291</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E72">A296</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E73">A297</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E74">A298</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">B</font> being  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c12">MD1</font>, <font color="Maroon" title="c13">MD2</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c11">B</font><b> such that </b><br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E75">A299</font></i>: 
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> &amp; <font color="Maroon" title="c12">MD1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c13">MD2</font> &amp; <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E32:32_3_1_1_1_3_1"><i><font color="Green" title="E70">A231</font></i></a>, <a class="ref" href="integra3.html#T14" target="_self" title="INTEGRA3:th.14">Th14</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E76">A300</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E77">A301</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font>
 ;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E78">A302</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><i><font color="Green" title="E79">A303</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E76">A300</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c9">j</font>

;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E80">A304</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_2"/><i><font color="Green" title="E81">A305</font></i>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c14">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_2"/><i><font color="Green" title="E82">A306</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_2"/><i><font color="Green" title="E83">A307</font></i>: 
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x1</font> &amp; <font color="Maroon" title="c15">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_2"><i><font color="Green" title="E81">A305</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_2"/>
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_2"/><b>then </b><i><font color="Green" title="E84">A308</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_2"><i><font color="Green" title="E82">A306</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> = 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E16:32_3_1_1_1_3_1"><i><font color="Green" title="E59">A220</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E77">A301</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E78">A302</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>

;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">x1</font> &amp; <font color="Maroon" title="c15">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_2"><i><font color="Green" title="E83">A307</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_2"><i><font color="Green" title="E84">A308</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_2"/>
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">B</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c11">B</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T38" title="FUNCT_2:th.38">FUNCT_2:38</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E81">A309</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c11">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_3"/><i><font color="Green" title="E82">A310</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">a</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_3"/><i><font color="Green" title="E83">A311</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Maroon" title="c15">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font></span>)</span> holds <br/><font color="Maroon" title="c15">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_3_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E84">A312</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E85">A313</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E80">A304</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c15">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_3"><i><font color="Green" title="E83">A311</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_3_1"><i><font color="Green" title="E85">A313</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_3_1"/>
<font color="Maroon" title="c15">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_3_1"><i><font color="Green" title="E84">A312</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green" title="E84">A314</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c11">B</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">a</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_3"/><i><font color="Green" title="E85">A315</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_3"><i><font color="Green" title="E82">A310</font></i></a>, <a class="ref" href="rfunct_1.html#D9" title="RFUNCT_1:def.9">RFUNCT_1:def 9</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font></span>)</span> holds <br/><font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_3_2"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_3_2"/><b>then </b><i><font color="Green" title="E86">A316</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_3_2"/><b>then </b><i><font color="Green" title="E87">A317</font></i>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E80">A304</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_3_2"/><b>then </b>
<font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">x</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_2_3"><i><font color="Green" title="E85">A315</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_3_2"><i><font color="Green" title="E87">A317</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_3_2"/>
<font color="Maroon" title="c14">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_3_2"><i><font color="Green" title="E86">A316</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c11">B</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D9" title="RFUNCT_1:def.9">RFUNCT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_3"/>
<font color="Maroon" title="c14">g</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c11">B</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_3"><i><font color="Green" title="E84">A314</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E82">A318</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E83">A319</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1"><i><font color="Green" title="E67">A228</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E84">A320</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E19:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E85">A321</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 ;<br/><a NAME="E20:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E86">A322</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1"/><i><font color="Green" title="E87">A323</font></i>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c15">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1"/><i><font color="Green" title="E88">A324</font></i>: 
<font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1"><i><font color="Green" title="E67">A228</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1"/>
<font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1"><i><font color="Green" title="E67">A228</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E16:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E82">A318</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E83">A319</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E85">A321</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E89">A325</font></i>: 
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E87">A323</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E88">A324</font></i></a>, <a class="ref" href="integra3.html#T15" target="_self" title="INTEGRA3:th.15">Th15</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1"/><i><font color="Green" title="E90">A326</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T28" title="JORDAN3:th.28">JORDAN3:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1"/>
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E87">A323</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E90">A326</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><i><font color="Green" title="E91">A327</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><i><font color="Green" title="E92">A328</font></i>: 
( <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E89">A325</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E91">A327</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E93">A329</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font> &amp; <font color="Maroon" title="c17">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E78">A302</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font>
 ;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E94">A330</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E92">A328</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E92">A328</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><i><font color="Green" title="E95">A331</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E93">A329</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E95">A331</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E96">A332</font></i>: 
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>then </b><font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n</font> = 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E93">A329</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E94">A330</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E95">A331</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font>

;<br/><b>hence </b><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E91">A327</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><i><font color="Green" title="E96">A332</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c17">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><i><font color="Green" title="E91">A333</font></i>: 
( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><i><font color="Green" title="E92">A334</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1"/>
( <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E74">A298</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E92">A334</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E89">A325</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1"><i><font color="Green" title="E53">A203</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E93">A335</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">n</font> or <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">n</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
not <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">n</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E94">A336</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E95">A337</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E94">A336</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_4_1"><i><font color="Green" title="E89">A325</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E95">A337</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E96">A338</font></i>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><i><font color="Green" title="E97">A339</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">y1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><i><font color="Green" title="E98">A340</font></i>: 
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
<font color="Maroon" title="c18">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E97">A339</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E98">A340</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green" title="E40">A20</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E94">A336</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E96">A338</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><i><font color="Green" title="E98">A340</font></i></a>, <a class="ref" href="integra3.html#T4" target="_self" title="INTEGRA3:th.4">Th4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E91">A333</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><i><font color="Green" title="E93">A335</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4"/><b>then </b><i><font color="Green" title="E87">A341</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2"/><i><font color="Green" title="E88">A342</font></i>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c15">x1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1"/>
( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> or <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_4_2"><i><font color="Green" title="E88">A342</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E89">A343</font></i>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T28" title="JORDAN3:th.28">JORDAN3:28</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E90">A344</font></i>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E89">A343</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E91">A345</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E92">A346</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E93">A347</font></i>: 
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E92">A346</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c9">j</font>

;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E91">A345</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E92">A346</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E94">A348</font></i>: 
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E89">A343</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E93">A347</font></i></a>, <a class="ref" href="integra3.html#T15" target="_self" title="INTEGRA3:th.15">Th15</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E95">A349</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E90">A344</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E96">A350</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E94">A348</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E95">A349</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E97">A351</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
<font color="Maroon" title="c17">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E96">A350</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E98">A352</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E20:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E96">A350</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E22:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E99">A353</font></i>: 
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E23:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green" title="E100">A354</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E84">A320</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E97">A351</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E98">A352</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E99">A353</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E24:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>then </b><font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n</font> = 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1"><i><font color="Green" title="E67">A228</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E97">A351</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E98">A352</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E99">A353</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font>
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E99">A353</font></i></a></i>
;<br/><b>hence </b><a NAME="E26:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E95">A349</font></i></a>, <a class="txt" href="integra3.html#E23:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><i><font color="Green" title="E100">A354</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b><i><font color="Green" title="E89">A355</font></i>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><i><font color="Green" title="E90">A356</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><i><font color="Green" title="E91">A357</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E89">A355</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E90">A356</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b><i><font color="Green" title="E92">A358</font></i>: 
( <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">x1</font> &amp; <font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E89">A355</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><i><font color="Green" title="E93">A359</font></i>: 
( <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">j1</font> = <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E72">A296</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E93">A359</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c17">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E91">A357</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E92">A358</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><i><font color="Green" title="E91">A357</font></i></a>, <a class="ref" href="integra3.html#T16" target="_self" title="INTEGRA3:th.16">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_2"/>
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_4"><i><font color="Green" title="E87">A341</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E87">A360</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E22:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E88">A361</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E23:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E89">A362</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_5_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/><i><font color="Green" title="E90">A363</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/><b>then </b><i><font color="Green" title="E91">A364</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E87">A360</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E79">A303</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_5_1_1"><i><font color="Green" title="E90">A363</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_5_1_1"><i><font color="Green" title="E91">A364</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><i><font color="Green" title="E90">A365</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b><i><font color="Green" title="E91">A366</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E87">A360</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><i><font color="Green" title="E92">A367</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E78">A302</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E87">A360</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_5_1_2"><i><font color="Green" title="E90">A365</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b><i><font color="Green" title="E93">A368</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b><i><font color="Green" title="E94">A369</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E76">A300</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_5_1_2"><i><font color="Green" title="E91">A366</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_5_1_2"><i><font color="Green" title="E92">A367</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_5_1_2"><i><font color="Green" title="E93">A368</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E76">A300</font></i></a>, <a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E79">A303</font></i></a>, <a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E88">A361</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_5_1_2"><i><font color="Green" title="E91">A366</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_5_1_2"><i><font color="Green" title="E94">A369</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:32_3_1_1_1_3_1_3_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E25:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E90">A370</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E23:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E89">A362</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E26:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E91">A371</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6"/>
(  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1"/><i><font color="Green" title="E92">A372</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c11">B</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6_1"/><i><font color="Green" title="E93">A373</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_6_1"><i><font color="Green" title="E93">A373</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_6_1"><i><font color="Green" title="E93">A373</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_6_1"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E23:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E89">A362</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_6_1"><i><font color="Green" title="E92">A372</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c11">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>

<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_6_1"/><b>then </b><i><font color="Green" title="E94">A374</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_6_1"/><i><font color="Green" title="E95">A375</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E72">A296</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
<font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_6_1"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_6_1"><i><font color="Green" title="E95">A375</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E7:32_3_1_1_1_3_1_3_1_1_2_6_1"><i><font color="Green" title="E94">A374</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E81">A309</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E86">A322</font></i></a>, <a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E90">A370</font></i></a>, <a class="ref" href="integra3.html#T12" target="_self" title="INTEGRA3:th.12">Th12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E28:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E92">A376</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/><a NAME="E29:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E93">A377</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/><a NAME="E30:32_3_1_1_1_3_1_3_1_1_2"/>
( <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R1" title="RFUNCT_1:pred.1">is_bounded_above_on</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/><a NAME="E31:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> )
 <b>by </b><i><a class="ref" href="integra1.html#T13" title="INTEGRA1:th.13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15" title="INTEGRA1:th.15">INTEGRA1:15</a></i>;<br/><a NAME="E32:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E28:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E92">A376</font></i></a>, <a class="txt" href="integra3.html#E29:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E93">A377</font></i></a>, <a class="ref" href="seq_4.html#T64" title="SEQ_4:th.64">SEQ_4:64</a>, <a class="ref" href="seq_4.html#T65" title="SEQ_4:th.65">SEQ_4:65</a></i>;<br/><a NAME="E33:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E94">A378</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E34:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="integra3.html#T8" target="_self" title="INTEGRA3:th.8">Th8</a></i>;<br/><a NAME="E35:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E33:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E94">A378</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E36:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E95">A379</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E91">A371</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E37:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c11">B</font></span>)</span>,<font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></i>;<br/><a NAME="E38:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c11">B</font></span>)</span>,<font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E40:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E96">A380</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c11">B</font></span>)</span>,<font color="Maroon" title="c12">MD1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c11">B</font></span>)</span>,<font color="Maroon" title="c12">MD1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E41:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c11">B</font></span>)</span>,<font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E40:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E96">A380</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E42:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E97">A381</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/><a NAME="E43:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E98">A382</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E40:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E96">A380</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/><a NAME="E44:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E45:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E46:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E99">A383</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> &amp; <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E42:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E97">A381</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E47:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E100">A384</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E48:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E101">A385</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E42:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E97">A381</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E49:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E78">A302</font></i></a>, <a class="txt" href="integra3.html#E46:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E99">A383</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E50:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E51:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E102">A386</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E47:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E100">A384</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E52:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E103">A387</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1"/>
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/><i><font color="Green" title="E104">A388</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E48:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E101">A385</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green" title="E105">A389</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E76">A300</font></i></a>, <a class="txt" href="integra3.html#E46:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E99">A383</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E46:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E99">A383</font></i></a>, <a class="txt" href="integra3.html#E51:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E102">A386</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"><i><font color="Green" title="E104">A388</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"><i><font color="Green" title="E105">A389</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><i><font color="Green" title="E104">A390</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E105">A391</font></i>: 
( <font color="Maroon" title="c15">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font> &amp; <font color="Maroon" title="c15">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E48:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E101">A385</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green" title="E106">A392</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c15">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c15">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E48:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E101">A385</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"><i><font color="Green" title="E104">A390</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E76">A300</font></i></a>, <a class="txt" href="integra3.html#E46:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E99">A383</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"><i><font color="Green" title="E105">A391</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"><i><font color="Green" title="E106">A392</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E46:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E99">A383</font></i></a>, <a class="txt" href="integra3.html#E51:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E102">A386</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_7"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_7"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E53:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E104">A393</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E51:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E102">A386</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E54:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E105">A394</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E43:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E98">A382</font></i></a>, <a class="txt" href="integra3.html#E52:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E103">A387</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/><a NAME="E55:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E53:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E104">A393</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/><a NAME="E56:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E57:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E54:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E105">A394</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E58:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E59:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E106">A395</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font>
 <b>by </b><i><a class="ref" href="integra1.html#D19" title="INTEGRA1:def.19">INTEGRA1:def 19</a></i>;<br/><a NAME="E60:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E12"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/><a NAME="E61:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E62:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E107">A396</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c12">MD1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E59:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E106">A395</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E63:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E108">A397</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E64:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E109">A398</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E65:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E73">A297</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E66:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E67:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E110">A399</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E68:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E111">A400</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E25:32_3_1_1_1_3_1"><i><font color="Green" title="E67">A228</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E69:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E112">A401</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 <b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/><a NAME="E70:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E113">A402</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_8"/><b>then </b><i><font color="Green" title="E114">A403</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E67:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E110">A399</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_8"/><i><font color="Green" title="E115">A404</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E64:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E109">A398</font></i></a>, <a class="txt" href="integra3.html#E68:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E111">A400</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_8"/><b>then </b><i><font color="Green" title="E116">A405</font></i>: 
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1"/>
( <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font> or <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E116">A405</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/><i><font color="Green" title="E117">A406</font></i>: 
( <font color="Maroon" title="c16">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E63:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E108">A397</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E115">A404</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"><i><font color="Green" title="E117">A406</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/><b>then </b><i><font color="Green" title="E118">A407</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E114">A403</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E115">A404</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"><i><font color="Green" title="E117">A406</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"><i><font color="Green" title="E117">A406</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"><i><font color="Green" title="E118">A407</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green" title="E117">A408</font></i>: 
( <font color="Maroon" title="c16">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green" title="E118">A409</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="txt" href="integra3.html#E63:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E108">A397</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E115">A404</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E117">A408</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green" title="E119">A410</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1"/>
( <font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E118">A409</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E119">A410</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1"><i><font color="Green" title="E53">A203</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E117">A408</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b><i><font color="Green" title="E120">A411</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green" title="E121">A412</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E21:32_3_1_1_1_3_1"><i><font color="Green" title="E64">A225</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E114">A403</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8"><i><font color="Green" title="E115">A404</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E117">A408</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green" title="E122">A413</font></i>: 
( <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E72">A296</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b><i><font color="Green" title="E123">A414</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E20:32_3_1_1_1_3_1"><i><font color="Green" title="E63">A224</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E122">A413</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green" title="E124">A415</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1"><i><font color="Green" title="E60">A221</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E123">A414</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E124">A415</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E121">A412</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1"><i><font color="Green" title="E49">A199</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E117">A408</font></i></a>, <a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><i><font color="Green" title="E120">A411</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E71:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green" title="E114">A416</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1"/><i><font color="Green" title="E115">A417</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E84">A320</font></i></a>, <a class="txt" href="integra3.html#E70:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E113">A402</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1"/><i><font color="Green" title="E116">A418</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1"/><i><font color="Green" title="E117">A419</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E67:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E110">A399</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E64:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E109">A398</font></i></a>, <a class="txt" href="integra3.html#E69:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E112">A401</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E116">A418</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1"/><b>then </b><i><font color="Green" title="E118">A420</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E115">A417</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E117">A419</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_9_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green" title="E119">A421</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E120">A422</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E121">A423</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">MD2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green" title="E122">A424</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green" title="E123">A425</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E64:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E109">A398</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E115">A417</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E116">A418</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E117">A419</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E120">A422</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E84">A320</font></i></a>, <a class="txt" href="integra3.html#E70:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E113">A402</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E124">A426</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E116">A418</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E125">A427</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green" title="E126">A428</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k2</font>
 ;<br/>
<a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E124">A426</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E125">A427</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E126">A428</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E127">A429</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E128">A430</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E123">A425</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E126">A428</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E126">A428</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/><b>then </b><i><font color="Green" title="E129">A431</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/><i><font color="Green" title="E130">A432</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E121">A423</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/><i><font color="Green" title="E131">A433</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E127">A429</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1"/>
( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/><i><font color="Green" title="E132">A434</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E130">A432</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E133">A435</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E64:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E109">A398</font></i></a>, <a class="txt" href="integra3.html#E70:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E113">A402</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E115">A417</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E120">A422</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E129">A431</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E131">A433</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E126">A428</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"><i><font color="Green" title="E132">A434</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"><i><font color="Green" title="E133">A435</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><i><font color="Green" title="E132">A436</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E133">A437</font></i>: 
( <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD2</font> &amp; <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E130">A432</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E134">A438</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/>
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E135">A439</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E115">A417</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">MD2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E130">A432</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"><i><font color="Green" title="E132">A436</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E64:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E109">A398</font></i></a>, <a class="txt" href="integra3.html#E70:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E113">A402</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E115">A417</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E119">A421</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E120">A422</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"><i><font color="Green" title="E133">A437</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"><i><font color="Green" title="E134">A438</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"><i><font color="Green" title="E135">A439</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E126">A428</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E129">A431</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><i><font color="Green" title="E131">A433</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green" title="E129">A440</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c13">MD2</font>,<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E122">A424</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">MD2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E121">A423</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E129">A440</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c6">D2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E121">A423</font></i></a>, <a class="txt" href="integra3.html#E16:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E128">A430</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2_9_1_1"><i><font color="Green" title="E129">A440</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_9_1"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_9_1"><i><font color="Green" title="E118">A420</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c13">MD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E72:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10"><b>proof </b></a><div class="add">

<i><font color="Green" title="E115">A441</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font>
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10"/><i><font color="Green" title="E116">A442</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10"/><i><font color="Green" title="E117">A443</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green" title="E118">A444</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E116">A442</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_10"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E119">A445</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green" title="E120">A446</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green" title="E121">A447</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">MD1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green" title="E122">A448</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E115">A441</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green" title="E123">A449</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E124">A450</font></i>: 
<font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k2</font>
 ;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E125">A451</font></i>: 
<font color="Maroon" title="c17">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E124">A450</font></i></a></i>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E126">A452</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E123">A449</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E124">A450</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green" title="E127">A453</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E126">A452</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green" title="E128">A454</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E115">A441</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E117">A443</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E120">A446</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E124">A450</font></i></a>, <a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E127">A453</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E124">A450</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green" title="E129">A455</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E130">A456</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E131">A457</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E121">A447</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E132">A458</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E127">A453</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green" title="E133">A459</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1"/>
( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c16">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/><i><font color="Green" title="E134">A460</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c11">B</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E131">A457</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n1</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E115">A441</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E120">A446</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E125">A451</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E129">A455</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E132">A458</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"><i><font color="Green" title="E134">A460</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><i><font color="Green" title="E134">A461</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E135">A462</font></i>: 
( <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font> &amp; <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E131">A457</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E136">A463</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/>
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T148" title="XREAL_1:th.148">XREAL_1:148</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E137">A464</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E115">A441</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">MD1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E17:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E131">A457</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"><i><font color="Green" title="E134">A461</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1"><i><font color="Green" title="E58">A219</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1"><i><font color="Green" title="E62">A223</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E75">A299</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E115">A441</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E119">A445</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E120">A446</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"><i><font color="Green" title="E135">A462</font></i></a>, <a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"><i><font color="Green" title="E136">A463</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"><i><font color="Green" title="E137">A464</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E124">A450</font></i></a>, <a class="txt" href="integra3.html#E15:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E129">A455</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E132">A458</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">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="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c12">MD1</font>,<font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E16:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E130">A456</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">MD1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E121">A447</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c17">k2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E133">A459</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<b>hence </b><a NAME="E22:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c16">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E122">A448</font></i></a>, <a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E128">A454</font></i></a>, <a class="txt" href="integra3.html#E19:32_3_1_1_1_3_1_3_1_1_2_10_3_1"><i><font color="Green" title="E133">A459</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p1">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E115">A441</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_3_1_1_2_10"><i><font color="Green" title="E118">A444</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_10"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c12">MD1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p2">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E73:32_3_1_1_1_3_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E36:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E95">A379</font></i></a>, <a class="txt" href="integra3.html#E62:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E107">A396</font></i></a>, <a class="txt" href="integra3.html#E71:32_3_1_1_1_3_1_3_1_1_2"><i><font color="Green" title="E114">A416</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E34:32_3_1_1_1_3_1"/><i><font color="Green" title="E72">A465</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_4"/><i><font color="Green" title="E73">A466</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E24:32_3_1_1_1_3_1"><i><font color="Green" title="E66">A227</font></i></a>, <a class="txt" href="integra3.html#E29:32_3_1_1_1_3_1"><i><font color="Green" title="E69">A230</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_4"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_3_1_1_1_3_1"><i><font color="Green" title="E65">A226</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_4"/><b>then </b><i><font color="Green" title="E74">A467</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_4"/><i><font color="Green" title="E75">A468</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_4"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:32_3_1_1_1_3_1"><i><font color="Green" title="E55">A205</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_4"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13:32_3_1_1_1_3_1"><i><font color="Green" title="E56">A206</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_4"><i><font color="Green" title="E73">A466</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1_4"><i><font color="Green" title="E75">A468</font></i></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E26:32_3_1_1_1_3_1"><i><font color="Green" title="E68">A229</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1_3_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1_3_1_4"><i><font color="Green" title="E74">A467</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:32_3_1_1_1_3_1"/><i><font color="Green" title="E73">A469</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_5"/><i><font color="Green" title="E74">A470</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_5"/><b>then </b><i><font color="Green" title="E75">A471</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_5"/>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_5"/><b>then </b>
<font color="Maroon" title="c10">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c10">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c10">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),1,<font color="Maroon" title="c9">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E14:32_3_1_1_1_3_1"><i><font color="Green" title="E57">A207</font></i></a>, <a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1_3_1_5"><i><font color="Green" title="E74">A470</font></i></a>, <a class="ref" href="integra2.html#T4" title="INTEGRA2:th.4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E18:32_3_1_1_1_3_1"><i><font color="Green" title="E61">A222</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_3_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1_3_1_5"><i><font color="Green" title="E75">A471</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:32_3_1_1_1_3_1"/><i><font color="Green" title="E74">A472</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_3_1_1_1_3_1"><i><font color="Green" title="E54">A204</font></i></a>, <a class="txt" href="integra3.html#E33:32_3_1_1_1_3_1"><i><font color="Green" title="E71">A232</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E37:32_3_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font>)</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">D2</font>),<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c10">n1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D1</font>),<span class="p4">(<span class="default"><font color="Maroon" title="c10">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c9">j</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E34:32_3_1_1_1_3_1"><i><font color="Green" title="E72">A465</font></i></a>, <a class="txt" href="integra3.html#E35:32_3_1_1_1_3_1"><i><font color="Green" title="E73">A469</font></i></a></i>;<br/>
<b>hence </b><a NAME="E38:32_3_1_1_1_3_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1_1_3_1"><i><font color="Green" title="E50">A200</font></i></a>, <a class="txt" href="integra3.html#E36:32_3_1_1_1_3_1"><i><font color="Green" title="E74">A472</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1"/><i><font color="Green" title="E47">A473</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="integra3.html#E3:32_3_1_1_1"><i><font color="Green" title="E45">A43</font></i></a>, <a class="txt" href="integra3.html#E4:32_3_1_1_1"><i><font color="Green" title="E46">A196</font></i></a>);<br/></i>
<a NAME="E6:32_3_1_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">i</font>]
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_3_1_1_1"><i><font color="Green" title="E47">A473</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Olive" title="b1">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green" title="E43">A23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:32_3_1_1"/><i><font color="Green" title="E43">A474</font></i>: 
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font> &amp; <font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green" title="E42">A22</font></i></a></i>;<br/>
<a NAME="E5:32_3_1_1"/><i><font color="Green" title="E44">A475</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E6:32_3_1_1"/><i><font color="Green" title="E45">A476</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_2"/><i><font color="Green" title="E46">A477</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 ;<br/>

<a NAME="E2:32_3_1_1_2"/><i><font color="Green" title="E47">A478</font></i>: 
<font color="Maroon" title="c4">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1"><i><font color="Green" title="E43">A474</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<a NAME="E3:32_3_1_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1"><i><font color="Green" title="E43">A474</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:32_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_2"><i><font color="Green" title="E46">A477</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:32_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1"><i><font color="Green" title="E43">A474</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1"><i><font color="Green" title="E44">A475</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E6:32_3_1_1_2"/><b>then </b><i><font color="Green" title="E48">A479</font></i>: 
<font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E7:32_3_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_2_1_1"/>
( <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1"><i><font color="Green" title="E43">A474</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_2"><i><font color="Green" title="E48">A479</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D1</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1"><i><font color="Green" title="E43">A474</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1_2"><i><font color="Green" title="E48">A479</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">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="E8:32_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_2"><i><font color="Green" title="E47">A478</font></i></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1"/><i><font color="Green" title="E46">A480</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E8:32_3_1_1"/><i><font color="Green" title="E47">A481</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="integra3.html#E5:32_3_1_1"><i><font color="Green" title="E44">A475</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E9:32_3_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E10:32_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E11:32_3_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c6">D2</font>,<font color="Maroon" title="c5">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_3_1_1"><i><font color="Green" title="E46">A480</font></i></a>, <a class="txt" href="integra3.html#E8:32_3_1_1"><i><font color="Green" title="E47">A481</font></i></a>, <a class="ref" href="goboard2.html#T19" title="GOBOARD2:th.19">GOBOARD2:19</a></i>;<br/>
<a NAME="E12:32_3_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">D2</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D2</font>) <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_3_1_1"><i><font color="Green" title="E43">A474</font></i></a>, <a class="txt" href="integra3.html#E6:32_3_1_1"><i><font color="Green" title="E45">A476</font></i></a>, <a class="ref" href="integra1.html#T45" title="INTEGRA1:th.45">INTEGRA1:45</a></i>;<br/>
<b>hence </b><a NAME="E13:32_3_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c6">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#T45" title="INTEGRA1:th.45">INTEGRA1:45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1"/>
 ex <font color="Olive" title="b1">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green" title="E41">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3"/>
 ex <font color="Olive" title="b1">D2</font> being   <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c4">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp; <font color="Maroon" title="c5">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Olive" title="b1">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c5">D1</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:32"/><i><font color="Green" title="E40">A482</font></i>: 
for <font color="Olive" title="b1">e</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">e</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:32_4"/><i><font color="Green" title="E41">A483</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:32_4"/><b>then </b><i><font color="Green" title="E42">A484</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c4">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:32_4"/><i><font color="Green" title="E43">A485</font></i>: 
 <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D16" title="INTEGRA1:def.16">INTEGRA1:def 16</a></i>;<br/>
<a NAME="E5:32_4"/><i><font color="Green" title="E44">A486</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="seq_4.html#V1" title="SEQ_4:attr.1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="integra2.html#T36" title="INTEGRA2:th.36">INTEGRA2:36</a></i>;<br/>
<a NAME="E6:32_4"/>
not  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E7:32_4"/><b>then </b>
not  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E9:32_4"/><i><font color="Green" title="E45">A487</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_4"><i><font color="Green" title="E42">A484</font></i></a>, <a class="txt" href="integra3.html#E4:32_4"><i><font color="Green" title="E43">A485</font></i></a>, <a class="txt" href="integra3.html#E5:32_4"><i><font color="Green" title="E44">A486</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">D</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E11:32_4"/><i><font color="Green" title="E46">A488</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="integra3.html#E9:32_4"><i><font color="Green" title="E45">A487</font></i></a>, <a class="txt" href="integra3.html#E18"><i><font color="Green" title="E18">Lm8</font></i></a></i>;<br/>
<a NAME="E12:32_4"/><i><font color="Green" title="E47">A489</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:32_4"><i><font color="Green" title="E46">A488</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">p</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font>;<br/>
<b>set </b><font color="Maroon" title="c9">H</font> =  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">h</font> =  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  =  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,$1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c11">v</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E14:32_4"/><i><font color="Green" title="E48">A490</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> holds <br/><font color="Maroon" title="c11">v</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c12">v1</font> being  <a href="integra2.html#V1" title="INTEGRA2:attr.1">non-decreasing</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E16:32_4"/><i><font color="Green" title="E49">A491</font></i>: 
<font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c12">v1</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="integra2.html#T3" title="INTEGRA2:th.3">INTEGRA2:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  );<br/>
<a NAME="E17:32_4"/><i><font color="Green" title="E50">A492</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c13">H</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:32_4_1"/><i><font color="Green" title="E51">A493</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font> &amp; <font color="Maroon" title="c13">H</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c13">H</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E16:32_4"><i><font color="Green" title="E49">A491</font></i></a>, <a class="ref" href="rfinseq.html#T3" title="RFINSEQ:th.3">RFINSEQ:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:32_4_1"/><i><font color="Green" title="E52">A494</font></i>: 
( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font> &amp;  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c14">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="integra3.html#T1" target="_self" title="INTEGRA3:th.1">Th1</a></i>;<br/>
<a NAME="E5:32_4_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_4_1"/><b>then </b>
( <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</font> &amp; <font color="Maroon" title="c11">v</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c14">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E4:32_4_1"><i><font color="Green" title="E52">A494</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:32_4_1"/><b>then </b><i><font color="Green" title="E53">Z</font></i>: 
( <font color="Maroon" title="c13">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_4_1"><i><font color="Green" title="E51">A493</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">Hk</font> = <font color="Maroon" title="c13">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E9:32_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">Hk</font>]
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32_4_1"><i><font color="Green" title="E53">Z</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:32_4_1"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c13">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:32_4"/><i><font color="Green" title="E51">A495</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">k</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="integra3.html#E17:32_4"><i><font color="Green" title="E50">A492</font></i></a>);<br/></i>
<a NAME="E20:32_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E21:32_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E22:32_4"/><b>then </b><i><font color="Green" title="E52">A496</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E23:32_4"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="integra3.html#E12"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<a NAME="E24:32_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E25:32_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E26:32_4"/><b>then </b><i><font color="Green" title="E53">A497</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_4"><i><font color="Green" title="E52">A496</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E27:32_4"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_4_2_1"/>
(  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> or  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_2_1_1"><b>suppose </b></a><a NAME="E1:32_4_2_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_4_2_1_1"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_2_1_2"><b>suppose </b></a><a NAME="E1:32_4_2_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_4_2_1_2"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4"><i><font color="Green" title="E41">A483</font></i></a>, <a class="txt" href="integra3.html#E26:32_4"><i><font color="Green" title="E53">A497</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c14">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E29:32_4"/><i><font color="Green" title="E54">A498</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/>
<a NAME="E30:32_4"/><i><font color="Green" title="E55">A499</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_4_3"/><i><font color="Green" title="E56">A500</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m</font>
 ;<br/>

<a NAME="E2:32_4_3"/><b>then </b><i><font color="Green" title="E57">A501</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E29:32_4"><i><font color="Green" title="E54">A498</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">D1</font> = <font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> ;<br/>
<b>consider </b><font color="Maroon" title="c17">D2</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E5:32_4_3"/><i><font color="Green" title="E58">A502</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c17">D2</font> &amp; <font color="Maroon" title="c16">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c17">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c17">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/>
<a NAME="E6:32_4_3"/>
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="rfunct_1.html#D11" title="RFUNCT_1:def.11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E7:32_4_3"/><b>then </b>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_4_3"><i><font color="Green" title="E58">A502</font></i></a>, <a class="ref" href="integra1.html#T48" title="INTEGRA1:th.48">INTEGRA1:48</a></i>;<br/>
<a NAME="E8:32_4_3"/><b>then </b><i><font color="Green" title="E59">A503</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/>
<a NAME="E9:32_4_3"/><i><font color="Green" title="E60">A504</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c16">D1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="ref" href="integra2.html#D3" title="INTEGRA2:def.3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E10:32_4_3"/><i><font color="Green" title="E61">A505</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c16">D1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_4_3"><i><font color="Green" title="E57">A501</font></i></a>, <a class="ref" href="integra2.html#D3" title="INTEGRA2:def.3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E11:32_4_3"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E12:32_4_3"/><b>then </b><i><font color="Green" title="E62">A506</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c16">D1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:32_4_3"><i><font color="Green" title="E61">A505</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:32_4_3"/><i><font color="Green" title="E63">A507</font></i>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_4_3_1"/><i><font color="Green" title="E64">A508</font></i>: 
for <font color="Olive" title="b1">n1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font> holds <br/> <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Olive" title="b1">n1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">n1</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_4_3_1_1"/><i><font color="Green" title="E65">A509</font></i>: 
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font>
 ;<br/>

<a NAME="E2:32_4_3_1_1"/><b>then </b><i><font color="Green" title="E66">A510</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">n1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_4_3_1_1_1"/>
( <font color="Maroon" title="c18">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="integra3.html#E2:32_4_3_1_1"><i><font color="Green" title="E66">A510</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_4_3_1_1_1_1"/><i><font color="Green" title="E67">A511</font></i>: 
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_4_3_1_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_1_1"><i><font color="Green" title="E65">A509</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_4_3_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/><b>hence </b><a NAME="E4:32_4_3_1_1_1_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_4"><i><font color="Green" title="E46">A488</font></i></a>, <a class="txt" href="integra3.html#E1:32_4_3_1_1_1_1"><i><font color="Green" title="E67">A511</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_4_3_1_1_1_2"/><i><font color="Green" title="E67">A512</font></i>: 
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_4_3_1_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_1_1"><i><font color="Green" title="E65">A509</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_4_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E68">A513</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/><a NAME="E4:32_4_3_1_1_1_2"/><i><font color="Green" title="E69">A514</font></i>: 
<font color="Maroon" title="c18">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_1_1"><i><font color="Green" title="E65">A509</font></i></a>, <a class="txt" href="integra3.html#E1:32_4_3_1_1_1_2"><i><font color="Green" title="E67">A512</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_4_3_1_1_1_2"/>
<font color="Maroon" title="c18">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E6:32_4_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E7:32_4_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_1_1"><i><font color="Green" title="E65">A509</font></i></a>, <a class="txt" href="integra3.html#E4:32_4_3_1_1_1_2"><i><font color="Green" title="E69">A514</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:32_4_3_1_1_1_2"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_4_3_1_1_1_2"><i><font color="Green" title="E68">A513</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:32_4_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:32_4_3_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:32_4_3_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:32_4_3_1"/><b>then </b><i><font color="Green" title="E65">A515</font></i>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_4_3_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:32_4"><i><font color="Green" title="E49">A491</font></i></a>, <a class="ref" href="rfinseq.html#T1" title="RFINSEQ:th.1">RFINSEQ:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:32_4_3_1"/><i><font color="Green" title="E66">A516</font></i>: 
( <font color="Maroon" title="c18">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">n1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32_4_3_1"><i><font color="Green" title="E65">A515</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E9:32_4_3_1"/><i><font color="Green" title="E67">A517</font></i>: 
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E8:32_4_3_1"><i><font color="Green" title="E66">A516</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:32_4_3_1"/><b>then </b><i><font color="Green" title="E68">A518</font></i>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">n1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E8:32_4_3_1"><i><font color="Green" title="E66">A516</font></i></a></i>;<br/>
<a NAME="E11:32_4_3_1"/>
<font color="Maroon" title="c18">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_4_3_1"><i><font color="Green" title="E67">A517</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:32_4_3_1"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_1"><i><font color="Green" title="E64">A508</font></i></a>, <a class="txt" href="integra3.html#E10:32_4_3_1"><i><font color="Green" title="E68">A518</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_4_3"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_4_3_2"/><i><font color="Green" title="E64">A519</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_4_3_2_1"/><i><font color="Green" title="E65">A520</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:32_4_3_2_1"/><i><font color="Green" title="E66">A521</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">v</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:32_4"><i><font color="Green" title="E49">A491</font></i></a>, <a class="ref" href="rfinseq.html#T16" title="RFINSEQ:th.16">RFINSEQ:16</a></i>;<br/>
<a NAME="E3:32_4_3_2_1"/><i><font color="Green" title="E67">A522</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_4_3_2_1"/>
<font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a>, <a class="txt" href="integra3.html#E2:32_4_3_2_1"><i><font color="Green" title="E66">A521</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_4_3_2_1"/><b>then </b><i><font color="Green" title="E68">A523</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:32_4_3_2_1"/><b>then </b><i><font color="Green" title="E69">A524</font></i>: 
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_2_1"><i><font color="Green" title="E65">A520</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:32_4_3_2_1"/>
<font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_4_3_2_1"><i><font color="Green" title="E67">A522</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:32_4_3_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_4_3_2_1"><i><font color="Green" title="E68">A523</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:32_4_3_2_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E10:32_4_3_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a>, <a class="txt" href="integra3.html#E13:32_4_3"><i><font color="Green" title="E63">A507</font></i></a>, <a class="txt" href="integra3.html#E6:32_4_3_2_1"><i><font color="Green" title="E69">A524</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_4_3_2"/>
(  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span> holds <br/> <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:32_4_3_2"/><i><font color="Green" title="E65">A525</font></i>: 
( <font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span> &amp;  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">m</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E5:32_4_3_2"/>
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_4_3_2"><i><font color="Green" title="E65">A525</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_4_3_2"/><b>then </b><i><font color="Green" title="E66">A526</font></i>: 
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E7:32_4_3_2"/><b>then </b>
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c18">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32_4_3_2"><i><font color="Green" title="E65">A525</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/>
<a NAME="E8:32_4_3_2"/><b>then </b><i><font color="Green" title="E67">A527</font></i>: 
<font color="Maroon" title="c11">v</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E6:32_4_3_2"><i><font color="Green" title="E66">A526</font></i></a></i>;<br/>
<a NAME="E9:32_4_3_2"/>
<font color="Maroon" title="c18">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E6:32_4_3_2"><i><font color="Green" title="E66">A526</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:32_4_3_2"/><b>then </b><i><font color="Green" title="E68">A528</font></i>: 
 <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c11">v</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_4_3_2"><i><font color="Green" title="E67">A527</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:32_4_3_2"/><i><font color="Green" title="E69">A529</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c11">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:32_4"><i><font color="Green" title="E49">A491</font></i></a>, <a class="ref" href="rfinseq.html#T1" title="RFINSEQ:th.1">RFINSEQ:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">m1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:32_4_3_2"/><i><font color="Green" title="E70">A530</font></i>: 
( <font color="Maroon" title="c19">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">v1</font> &amp;  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c19">m1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E10:32_4_3_2"><i><font color="Green" title="E68">A528</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E14:32_4_3_2"/>
<font color="Maroon" title="c19">m1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E13:32_4_3_2"><i><font color="Green" title="E70">A530</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:32_4_3_2"/><b>then </b><i><font color="Green" title="E71">A531</font></i>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a>, <a class="txt" href="integra3.html#E1:32_4_3_2"><i><font color="Green" title="E64">A519</font></i></a>, <a class="txt" href="integra3.html#E13:32_4_3_2"><i><font color="Green" title="E70">A530</font></i></a>, <a class="ref" href="integra2.html#T2" title="INTEGRA2:th.2">INTEGRA2:2</a></i>;<br/>
<a NAME="E16:32_4_3_2"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c12">v1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4"><i><font color="Green" title="E51">A495</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">k2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E18:32_4_3_2"/><i><font color="Green" title="E72">A532</font></i>: 
( <font color="Maroon" title="c20">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">v</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">v</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">k2</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E11:32_4_3_2"><i><font color="Green" title="E69">A529</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E19:32_4_3_2"/><i><font color="Green" title="E73">A533</font></i>: 
<font color="Maroon" title="c20">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E18:32_4_3_2"><i><font color="Green" title="E72">A532</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:32_4_3_2"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c20">k2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:32_4"><i><font color="Green" title="E48">A490</font></i></a>, <a class="txt" href="integra3.html#E18:32_4_3_2"><i><font color="Green" title="E72">A532</font></i></a></i>;<br/>
<a NAME="E21:32_4_3_2"/><b>then </b><i><font color="Green" title="E74">A534</font></i>: 
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c20">k2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4_3_2"><i><font color="Green" title="E73">A533</font></i></a>, <a class="ref" href="integra1.html#T22" title="INTEGRA1:th.22">INTEGRA1:22</a></i>;<br/>
<a NAME="E22:32_4_3_2"/>
<font color="Maroon" title="c20">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:32_4_3_2"><i><font color="Green" title="E73">A533</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E23:32_4_3_2"/><b>then </b>
<font color="Maroon" title="c20">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:32_4_3_2"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E21:32_4_3_2"><i><font color="Green" title="E74">A534</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:32_4_3_2"/><b>then </b>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4" title="PRE_CIRC:def.4">PRE_CIRC:def 4</a></i>;<br/>
<b>hence </b><a NAME="E26:32_4_3_2"/>
<font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K3" title="SEQ_4:func.3">min</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5" title="RFUNCT_1:func.5">chi</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span>)</span>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_4_3_2"><i><font color="Green" title="E64">A519</font></i></a>, <a class="txt" href="integra3.html#E15:32_4_3_2"><i><font color="Green" title="E71">A531</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c18">D3</font> being    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E16:32_4_3"/><i><font color="Green" title="E64">A535</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c18">D3</font> &amp; <font color="Maroon" title="c16">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c18">D3</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c18">D3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c16">D1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c18">D3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c16">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green" title="E39">A19</font></i></a>, <a class="txt" href="integra3.html#E12:32_4_3"><i><font color="Green" title="E62">A506</font></i></a></i>;<br/>
<a NAME="E17:32_4_3"/><i><font color="Green" title="E65">A536</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c16">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_4_3"><i><font color="Green" title="E58">A502</font></i></a>, <a class="txt" href="integra3.html#E16:32_4_3"><i><font color="Green" title="E64">A535</font></i></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></i>;<br/>
<a NAME="E18:32_4_3"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<a NAME="E19:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E20:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_4_3"><i><font color="Green" title="E57">A501</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E21:32_4_3"/><b>then </b><i><font color="Green" title="E66">A537</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_4_3"><i><font color="Green" title="E60">A504</font></i></a>, <a class="txt" href="integra3.html#E17:32_4_3"><i><font color="Green" title="E65">A536</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:32_4_3"/><i><font color="Green" title="E67">A538</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E29:32_4"><i><font color="Green" title="E54">A498</font></i></a>, <a class="txt" href="integra3.html#E1:32_4_3"><i><font color="Green" title="E56">A500</font></i></a></i>;<br/>
<a NAME="E23:32_4_3"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">v1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c13">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E24:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E22:32_4_3"><i><font color="Green" title="E67">A538</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E25:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E26:32_4"><i><font color="Green" title="E53">A497</font></i></a>, <a class="ref" href="real_2.html#T177" title="REAL_2:th.177">REAL_2:177</a></i>;<br/>
<a NAME="E26:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
;<br/>
<a NAME="E27:32_4_3"/><b>then </b><i><font color="Green" title="E68">A539</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="ref" href="real_2.html#T178" title="REAL_2:th.178">REAL_2:178</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">s</font> =  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c20">sD</font> =  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font>;<br/>
<b>set </b><font color="Maroon" title="c21">sD1</font> =  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c22">sD2</font> =  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font>;<br/>
<a NAME="E28:32_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:32_4"><i><font color="Green" title="E45">A487</font></i></a>, <a class="txt" href="integra3.html#E12:32_4"><i><font color="Green" title="E47">A489</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E29:32_4_3"/><b>then </b><i><font color="Green" title="E69">A540</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E30:32_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span>
 
;<br/>
<a NAME="E31:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c17">D2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32_4_3"><i><font color="Green" title="E59">A503</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E32:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2" title="INTEGRA2:func.2">delta</a> <font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E21:32_4_3"><i><font color="Green" title="E66">A537</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E27:32_4_3"><i><font color="Green" title="E68">A539</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E35:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E29:32_4_3"><i><font color="Green" title="E69">A540</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:32_4_3"/><b>then </b><i><font color="Green" title="E70">A541</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="ref" href="integra2.html#D5" title="INTEGRA2:def.5">INTEGRA2:def 5</a></i>;<br/>
<a NAME="E37:32_4_3"/>
<font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font>
 
;<br/>
<a NAME="E38:32_4_3"/><b>then </b><i><font color="Green" title="E71">A542</font></i>: 
<font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E39:32_4_3"/>
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra2.html#D5" title="INTEGRA2:def.5">INTEGRA2:def 5</a></i>;<br/>
<a NAME="E40:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E38:32_4_3"><i><font color="Green" title="E71">A542</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E41:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E38:32_4_3"><i><font color="Green" title="E71">A542</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E42:32_4_3"/><b>then </b>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32_4"><i><font color="Green" title="E44">A486</font></i></a>, <a class="ref" href="seq_4.html#D4" title="SEQ_4:def.4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E43:32_4_3"/><b>then </b>
 <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D16" title="INTEGRA1:def.16">INTEGRA1:def 16</a></i>;<br/>
<a NAME="E44:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E45:32_4_3"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E36:32_4_3"><i><font color="Green" title="E70">A541</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E46:32_4_3"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138" title="COMPLEX1:th.138">COMPLEX1:138</a></i>;<br/>
<b>hence </b><a NAME="E47:32_4_3"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c15">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">e</font>
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c14">n</font>
;<br/>

<b>thus </b><a NAME="E31:32_4"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">e</font>
 <b>by </b><i><a class="txt" href="integra3.html#E30:32_4"><i><font color="Green" title="E55">A499</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:32"/>
 <a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="ref" href="seq_2.html#D6" title="SEQ_2:def.6">SEQ_2:def 6</a></i>;<br/>

<b>hence </b><a NAME="E11:32"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4" title="INTEGRA2:func.4">lower_sum</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K12" title="INTEGRA1:func.12">lower_integral</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="integra3.html#E9:32"><i><font color="Green" title="E40">A482</font></i></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</a></i>;<br/>


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