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

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c3">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">D1</font>, <font color="Maroon" title="c5">D2</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="c2">A</font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">x</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="c4">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="c4">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 ;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c4">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c5">D2</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">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="c4">D1</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"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 ;<br/>

<b>assume </b><a NAME="E3:17"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c3">g</font> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<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="c2">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</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="c2">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,$1;<br/>
<b>set </b><font color="Maroon" title="c6">j</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font>;<br/>
<a NAME="E4:17"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<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="E5:17"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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">D1</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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">D2</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="E6:17"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="c4">D1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="c5">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="E7:17"/><i><font color="Green" title="E23">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E8:17"/><i><font color="Green" title="E24">A7</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="c3">g</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="c3">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:17"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="integra3.html#E12"><i><font color="Green" title="E12">Lm4</font></i></a></i>;<br/>
<a NAME="E9:17"/><i><font color="Green" title="E25">A8</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">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="c5">D2</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E10:17"/><b>then </b><i><font color="Green" title="E26">A9</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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">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="E11:17"/><i><font color="Green" title="E27">A10</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">D2</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E3:17_1"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E9:17"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E4:17_1"/><i><font color="Green" title="E29">A12</font></i>: 
not <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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> <font color="Maroon" title="c4">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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> <font color="Maroon" title="c4">D1</font>
 ;<br/>

<a NAME="E2:17_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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="E3:17_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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="c2">A</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17_1_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E4:17_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E3:17_1"><i><font color="Green" title="E28">A11</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="E5:17_1"/>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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> <font color="Maroon" title="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</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="E6:17_1"/><b>then </b>
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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> <font color="Maroon" title="c4">D1</font> or <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span> <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"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:17_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:17_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:17_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</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="c4">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="c4">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E9:17_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E7:17"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="integra3.html#E3:17_1"><i><font color="Green" title="E28">A11</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="E12:17"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span></span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E10:17"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E13:17"/><i><font color="Green" title="E29">A15</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E14:17"/><i><font color="Green" title="E30">A16</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="c4">D1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E7:17"><i><font color="Green" title="E23">A6</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="c7">j1</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E7:17"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>
<a NAME="E16:17"/><i><font color="Green" title="E31">A17</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:17_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>

<b>hence </b><a NAME="E2:17_2"/>
( 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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">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="E17:17"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c5">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="E18:17"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>;<br/>
<a NAME="E19:17"/><i><font color="Green" title="E33">A19</font></i>: 
( <font color="Maroon" title="c7">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="c4">D1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font> &amp; <font color="Maroon" title="c7">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="c4">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E20:17"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,<font color="Maroon" title="c7">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="E21:17"/><b>then </b><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font> is <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>;<br/>
<a NAME="E22:17"/><i><font color="Green" title="E35">A21</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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E15"><i><font color="Green" title="E15">Lm7</font></i></a></i>;<br/>
<a NAME="E23:17"/><b>then </b><i><font color="Green" title="E36">A22</font></i>: 
<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:17"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="integra3.html#E21:17"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></i>;<br/>
<a NAME="E24:17"/><i><font color="Green" title="E37">A23</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="c7">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:17_3"/><i><font color="Green" title="E38">A24</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font> )
 ;<br/>

<b>assume </b><a NAME="E2:17_3"/><i><font color="Green" title="E39">A25</font></i>: 
<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_3_1_1"/>
( <font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> or <font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E2:17_3"><i><font color="Green" title="E39">A25</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="17_3_1_1_1"><b>suppose </b></a><a NAME="E1:17_3_1_1_1"/><i><font color="Green" title="E40">A26</font></i>: 
<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><a NAME="E2:17_3_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">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="c4">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:17_3_1_1_1"/><b>then </b><i><font color="Green" title="E41">A27</font></i>: 
<font color="Maroon" title="c8">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="c4">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="E4:17_3_1_1_1"/><b>then </b><i><font color="Green" title="E42">A28</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/><a NAME="E5:17_3_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">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="E6:17_3_1_1_1"/><b>then </b><i><font color="Green" title="E43">A29</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="integra3.html#E3:17_3_1_1_1"><i><font color="Green" title="E41">A27</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="E7:17_3_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="E8:17_3_1_1_1"/><b>then </b><i><font color="Green" title="E44">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/><a NAME="E9:17_3_1_1_1"/><i><font color="Green" title="E45">A31</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="integra3.html#E1:17_3_1_1_1"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:17_3_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:17_3_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:17_3_1_1_1"><i><font color="Green" title="E43">A29</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E12:17_3_1_1_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</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="E13:17_3_1_1_1"/><b>then </b><i><font color="Green" title="E46">A32</font></i>: 
<font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3_1_1_1"><i><font color="Green" title="E40">A26</font></i></a>, <a class="txt" href="integra3.html#E3:17_3_1_1_1"><i><font color="Green" title="E41">A27</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E14:17_3_1_1_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c7">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:17_3_1_1_1"><i><font color="Green" title="E43">A29</font></i></a>, <a class="txt" href="integra3.html#E9:17_3_1_1_1"><i><font color="Green" title="E45">A31</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="E15:17_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="integra3.html#E23:17"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="integra3.html#E4:17_3_1_1_1"><i><font color="Green" title="E42">A28</font></i></a>, <a class="txt" href="integra3.html#E8:17_3_1_1_1"><i><font color="Green" title="E44">A30</font></i></a>, <a class="txt" href="integra3.html#E13:17_3_1_1_1"><i><font color="Green" title="E46">A32</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="17_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_3_1_1_2"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><a NAME="E2:17_3_1_1_2"/>
<font color="Maroon" title="c8">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="c7">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:17_3_1_1_2"/><b>then </b><i><font color="Green" title="E41">A34</font></i>: 
<font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/><a NAME="E4:17_3_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">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="c4">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E5:17_3_1_1_2"/><b>then </b><i><font color="Green" title="E42">A35</font></i>: 
<font color="Maroon" title="c8">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="c4">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="E6:17_3_1_1_2"/><b>then </b><i><font color="Green" title="E43">A36</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/><a NAME="E7:17_3_1_1_2"/>
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="integra3.html#E5:17_3_1_1_2"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="integra3.html#T6" target="_self" title="INTEGRA3:th.6">Th6</a></i>;<br/><a NAME="E8:17_3_1_1_2"/><b>then </b><i><font color="Green" title="E44">A37</font></i>: 
<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3_1_1_2"><i><font color="Green" title="E40">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="E9:17_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">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">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:17_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">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">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:17_3_1_1_2"/><b>then </b><i><font color="Green" title="E45">A38</font></i>: 
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3_1_1_2"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="integra3.html#E6:17_3_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/><a NAME="E12:17_3_1_1_2"/>
<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_3"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="integra3.html#E8:17_3_1_1_2"><i><font color="Green" title="E44">A37</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="E13:17_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="integra3.html#E23:17"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="integra3.html#E3:17_3_1_1_2"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="integra3.html#E6:17_3_1_1_2"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="integra3.html#E11:17_3_1_1_2"><i><font color="Green" title="E45">A38</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="E4:17_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E25:17"/><i><font color="Green" title="E38">A39</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:17"><i><font color="Green" title="E32">A18</font></i></a>, <a class="txt" href="integra3.html#E21:17"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="integra3.html#E22:17"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="integra3.html#T5" target="_self" title="INTEGRA3:th.5">Th5</a></i>;<br/>
<a NAME="E26:17"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E27:17"/><b>then </b><i><font color="Green" title="E39">A40</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E25:17"><i><font color="Green" title="E38">A39</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E28:17"/>
<font color="Maroon" title="c7">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="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</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="E29:17"/><b>then </b>
<font color="Maroon" title="c7">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="c4">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="E30:17"/><b>then </b>
<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</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="E31:17"/><b>then </b><i><font color="Green" title="E40">A41</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E27:17"><i><font color="Green" title="E39">A40</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E32:17"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E33:17"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">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="E34:17"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</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="E35:17"/><b>then </b><i><font color="Green" title="E41">A42</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E31:17"><i><font color="Green" title="E40">A41</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E36:17"/>
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>  <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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:17_4"/><i><font color="Green" title="E42">A43</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> )
 ;<br/>

<a NAME="E2:17_4"/><i><font color="Green" title="E43">A44</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">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="c4">D1</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E3:17_4"/><b>then </b><i><font color="Green" title="E44">A45</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E1:17_4"><i><font color="Green" title="E42">A43</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:17_4"/><b>then </b><i><font color="Green" title="E45">A46</font></i>: 
<font color="Maroon" title="c8">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="c7">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="E5:17_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">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="c4">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E3:17_4"><i><font color="Green" title="E44">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:17_4"/><b>then </b><i><font color="Green" title="E46">A47</font></i>: 
<font color="Maroon" title="c8">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="c4">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="E7:17_4"/><i><font color="Green" title="E47">A48</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_4_1"/><i><font color="Green" title="E48">A49</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font>,<font color="Maroon" title="c8">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="E2:17_4_1"/><i><font color="Green" title="E49">A50</font></i>: 
<font color="Maroon" title="c8">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="c4">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17_4"><i><font color="Green" title="E46">A47</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="E3:17_4_1"/><b>then </b><i><font color="Green" title="E50">A51</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E4:17_4_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="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_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="E1:17_4_1_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">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="17_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_4_1_1_1_1"/><i><font color="Green" title="E51">A52</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:17_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E52">A53</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">D1</font>,<font color="Maroon" title="c8">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="c2">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">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17_4_1"><i><font color="Green" title="E49">A50</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:17_4_1_1_1_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E24:17"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="integra3.html#E1:17_4_1_1_1_1"><i><font color="Green" title="E51">A52</font></i></a></i>;<br/><b>hence </b><a NAME="E4:17_4_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="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:17_4_1"><i><font color="Green" title="E50">A51</font></i></a>, <a class="txt" href="integra3.html#E2:17_4_1_1_1_1"><i><font color="Green" title="E52">A53</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="17_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_4_1_1_1_2"/><i><font color="Green" title="E51">A54</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:17_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E52">A55</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">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17_4_1"><i><font color="Green" title="E49">A50</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:17_4_1_1_1_2"/><i><font color="Green" title="E53">A56</font></i>: 
( <font color="Maroon" title="c8">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="c4">D1</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">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#E2:17_4_1"><i><font color="Green" title="E49">A50</font></i></a>, <a class="txt" href="integra3.html#E1:17_4_1_1_1_2"><i><font color="Green" title="E51">A54</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">k1</font> = <font color="Maroon" title="c8">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#E2:17_4_1"><i><font color="Green" title="E49">A50</font></i></a>, <a class="txt" href="integra3.html#E1:17_4_1_1_1_2"><i><font color="Green" title="E51">A54</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E5:17_4_1_1_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E6:17_4_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">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="E7:17_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E54">A57</font></i>: 
<font color="Maroon" title="c9">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E3:17_4"><i><font color="Green" title="E44">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:17_4_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E3:17_4_1_1_1_2"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:17_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E55">A58</font></i>: 
<font color="Maroon" title="c9">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c9">k1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E24:17"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="integra3.html#E7:17_4_1_1_1_2"><i><font color="Green" title="E54">A57</font></i></a></i>;<br/><a NAME="E10:17_4_1_1_1_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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#E24:17"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="integra3.html#E3:17_4"><i><font color="Green" title="E44">A45</font></i></a>, <a class="txt" href="integra3.html#E1:17_4_1_1_1_2"><i><font color="Green" title="E51">A54</font></i></a></i>;<br/><a NAME="E11:17_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E56">A59</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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</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="c5">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:17_4_1"><i><font color="Green" title="E50">A51</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E12:17_4_1_1_1_2"/>
<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c9">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E24:17"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="integra3.html#E3:17_4"><i><font color="Green" title="E44">A45</font></i></a>, <a class="txt" href="integra3.html#E9:17_4_1_1_1_2"><i><font color="Green" title="E55">A58</font></i></a></i>;<br/><b>hence </b><a NAME="E13:17_4_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="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E2:17_4_1"><i><font color="Green" title="E49">A50</font></i></a>, <a class="txt" href="integra3.html#E2:17_4_1_1_1_2"><i><font color="Green" title="E52">A55</font></i></a>, <a class="txt" href="integra3.html#E3:17_4_1_1_1_2"><i><font color="Green" title="E53">A56</font></i></a>, <a class="txt" href="integra3.html#E11:17_4_1_1_1_2"><i><font color="Green" title="E56">A59</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="E5:17_4_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_4_1"><i><font color="Green" title="E48">A49</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="E8:17_4"/>
<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="integra3.html#E2:17_4"><i><font color="Green" title="E43">A44</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="E9:17_4"/><b>then </b>
<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">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="E48">A60</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="integra3.html#E4:17_4"><i><font color="Green" title="E45">A46</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#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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c5">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:17_4"><i><font color="Green" title="E46">A47</font></i></a>, <a class="txt" href="integra3.html#E7:17_4"><i><font color="Green" title="E47">A48</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>
;<br/>
<a NAME="E11:17_4"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c7">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E24:17"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="integra3.html#E3:17_4"><i><font color="Green" title="E44">A45</font></i></a>, <a class="txt" href="integra3.html#E4:17_4"><i><font color="Green" title="E45">A46</font></i></a></i>;<br/>
<a NAME="E12:17_4"/><b>then </b><i><font color="Green" title="E49">A61</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E24:17"><i><font color="Green" title="E37">A23</font></i></a></i>;<br/>
<a NAME="E13:17_4"/><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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="E14:17_4"/><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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:17_4"/><b>then </b><i><font color="Green" title="E50">A62</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">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">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="E16:17_4"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</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="E17:17_4"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</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="E18:17_4"/><b>then </b><i><font color="Green" title="E51">A63</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E24:17"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="integra3.html#E3:17_4"><i><font color="Green" title="E44">A45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E12:17_4"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="integra3.html#E18:17_4"><i><font color="Green" title="E51">A63</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#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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c5">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E15:17_4"><i><font color="Green" title="E50">A62</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E20:17_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="integra3.html#E10:17_4"><i><font color="Green" title="E48">A60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E37:17"/><b>then </b><i><font color="Green" title="E42">A64</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E35:17"><i><font color="Green" title="E41">A42</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E38:17"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</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="E39:17"/><b>then </b><i><font color="Green" title="E43">A65</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</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="E40:17"/>
<font color="Maroon" title="c7">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="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</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="E41:17"/><b>then </b><i><font color="Green" title="E44">A66</font></i>: 
<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</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/>
<i><font color="Green" title="E45">A67</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E39:17"><i><font color="Green" title="E43">A65</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="c4">D1</font>,<font color="Maroon" title="c7">j1</font>)
<b>by </b><i><a class="txt" href="integra3.html#E37:17"><i><font color="Green" title="E42">A64</font></i></a>, <a class="txt" href="integra3.html#E41:17"><i><font color="Green" title="E44">A66</font></i></a>, <a class="ref" href="integra1.html#D22" title="INTEGRA1:def.22">INTEGRA1:def 22</a></i>
;<br/>
<a NAME="E43:17"/><i><font color="Green" title="E46">A68</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><b>proof </b></a><div class="add">

<a NAME="E1:17_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E2:17_6"/><b>then </b>
<font color="Maroon" title="c7">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="c4">D1</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="E3:17_6"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</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="E4:17_6"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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/>


</div><b>end;</b></div>
<a NAME="E44:17"/><i><font color="Green" title="E47">A69</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><b>proof </b></a><div class="add">

<a NAME="E1:17_7"/><i><font color="Green" title="E48">A70</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:17_7_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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">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="E3:17_7_1"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</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/>
<b>hence </b><a NAME="E4:17_7_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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="E2:17_7"/><b>then </b><i><font color="Green" title="E49">A71</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:17_7"/><i><font color="Green" title="E50">A72</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="17_7_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_7_2"/><i><font color="Green" title="E51">A73</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c8">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c9">ID2</font> = <font color="Maroon" title="c8">ID1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E4:17_7_2"/><i><font color="Green" title="E52">A74</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">ID1</font> &amp; <font color="Maroon" title="c8">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">ID2</font> &amp; <font color="Maroon" title="c9">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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:17_7_2_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">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="E2:17_7_2_1"/>
<font color="Maroon" title="c8">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">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="E3:17_7_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17_7_2"><i><font color="Green" title="E51">A73</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="E4:17_7_2_1"/>
<font color="Maroon" title="c9">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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_7_2"/><i><font color="Green" title="E53">A75</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E6:17_7_2"/><b>then </b><i><font color="Green" title="E54">A76</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">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="E7:17_7_2"/><i><font color="Green" title="E55">A77</font></i>: 
<font color="Maroon" title="c8">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="c5">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_2_2"/><i><font color="Green" title="E56">A78</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">ID1</font> &amp; <font color="Maroon" title="c8">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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="integra3.html#E4:17_7_2"><i><font color="Green" title="E52">A74</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:17_7_2_2"/><b>then </b>
<font color="Maroon" title="c8">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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17_7_2"><i><font color="Green" title="E54">A76</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="E3:17_7_2_2"/>
<font color="Maroon" title="c8">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="c5">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_7_2_2"><i><font color="Green" title="E56">A78</font></i></a>, <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="E8:17_7_2"/><i><font color="Green" title="E56">A79</font></i>: 
<font color="Maroon" title="c9">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="c5">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_2_3"/><i><font color="Green" title="E57">A80</font></i>: 
(  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">ID2</font> &amp; <font color="Maroon" title="c9">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="c5">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:17_7_2"><i><font color="Green" title="E52">A74</font></i></a>, <a class="txt" href="integra3.html#E6:17_7_2"><i><font color="Green" title="E54">A76</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:17_7_2_3"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">ID2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</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="E3:17_7_2_3"/>
<font color="Maroon" title="c9">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="c5">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_7_2_3"><i><font color="Green" title="E57">A80</font></i></a>, <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="E9:17_7_2"/><b>then </b><i><font color="Green" title="E57">A81</font></i>: 
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="integra3.html#E4:17_7_2"><i><font color="Green" title="E52">A74</font></i></a>, <a class="txt" href="integra3.html#E5:17_7_2"><i><font color="Green" title="E53">A75</font></i></a>, <a class="txt" href="integra3.html#E7:17_7_2"><i><font color="Green" title="E55">A77</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E10:17_7_2"/><i><font color="Green" title="E58">A82</font></i>: 
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</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"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</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"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_2_4"/><i><font color="Green" title="E59">A83</font></i>: 
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">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">D2</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">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">D2</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:17_7_2"><i><font color="Green" title="E55">A77</font></i></a>, <a class="txt" href="integra3.html#E8:17_7_2"><i><font color="Green" title="E56">A79</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="E2:17_7_2_4"/><i><font color="Green" title="E60">A84</font></i>: 
( <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> &amp; <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E3:17_7_2_4"/>
( not <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">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">D1</font> &amp; not <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">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">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_7_2_4_1"/><i><font color="Green" title="E61">A85</font></i>: 
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">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">D1</font> or <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">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">D1</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_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="E1:17_7_2_4_1_1_1"/>
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">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">D1</font> or <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">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">D1</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E1:17_7_2_4_1"><i><font color="Green" title="E61">A85</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_2_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_7_2_4_1_1_1_1"/>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">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">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">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="E3:17_7_2_4_1_1_1_1"/><i><font color="Green" title="E62">A86</font></i>: 
( <font color="Maroon" title="c10">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">D1</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:17_7_2_4_1_1_1_1"/>
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:17_7_2"><i><font color="Green" title="E57">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="E5:17_7_2_4_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">n</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">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_2_4"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="integra3.html#E3:17_7_2_4_1_1_1_1"><i><font color="Green" title="E62">A86</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E6:17_7_2_4_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">D1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">n</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">D1</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="E7:17_7_2_4_1_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="17_7_2_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_7_2_4_1_1_1_2"/>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">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">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">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="E3:17_7_2_4_1_1_1_2"/><i><font color="Green" title="E62">A87</font></i>: 
( <font color="Maroon" title="c10">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">D1</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/><a NAME="E4:17_7_2_4_1_1_1_2"/>
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</font> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E9:17_7_2"><i><font color="Green" title="E57">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="E5:17_7_2_4_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c7">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">n</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">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_2_4"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="integra3.html#E3:17_7_2_4_1_1_1_2"><i><font color="Green" title="E62">A87</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E6:17_7_2_4_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">D1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c10">n</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">D1</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="E7:17_7_2_4_1_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="E3:17_7_2_4_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_7_2_4"/>
( <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</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"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</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"><font color="Maroon" title="c1">x</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E1:17_7_2_4"><i><font color="Green" title="E59">A83</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="E11:17_7_2"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:17_7_2"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c8">ID1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c9">ID2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:17_7_2"><i><font color="Green" title="E58">A82</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="E13:17_7_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E4:17_7_2"><i><font color="Green" title="E52">A74</font></i></a>, <a class="txt" href="integra3.html#E7:17_7_2"><i><font color="Green" title="E55">A77</font></i></a>, <a class="txt" href="integra3.html#E8:17_7_2"><i><font color="Green" title="E56">A79</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="E4:17_7"/>
( 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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="17_7_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_3"/><i><font color="Green" title="E51">A88</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E2:17_7_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E3:17_7_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ;<br/>

<b>thus </b><a NAME="E4:17_7_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#E3:17_7"><i><font color="Green" title="E50">A72</font></i></a>, <a class="txt" href="integra3.html#E1:17_7_3"><i><font color="Green" title="E51">A88</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:17_7"/><b>then </b><i><font color="Green" title="E51">A89</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E6:17_7"/><i><font color="Green" title="E52">A90</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="c4">D1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E7:17_7"/>
<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">D1</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E8:17_7"/><b>then </b><i><font color="Green" title="E53">A91</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17_7"><i><font color="Green" title="E52">A90</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E9:17_7"/>
<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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17_7"><i><font color="Green" title="E52">A90</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E10:17_7"/><b>then </b>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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#E8:17_7"><i><font color="Green" title="E53">A91</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E11:17_7"/><b>then </b><i><font color="Green" title="E54">A92</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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="E12:17_7"/><i><font color="Green" title="E55">A93</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">D2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_4"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</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="E2:17_7_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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#E5:17_7"><i><font color="Green" title="E51">A89</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="17_7_5_1_1"><b>suppose </b></a><a NAME="E1:17_7_5_1_1"/><i><font color="Green" title="E56">A94</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:17_7_5_1_1"/><i><font color="Green" title="E57">A95</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_1_1_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="txt" href="integra3.html#E1:17_7_5_1_1"><i><font color="Green" title="E56">A94</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:17_7_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:17_7_5_1_1"/><i><font color="Green" title="E58">A96</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></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="c4">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="c4">D1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E7:17"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E2:17_7_5_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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E3:17_7_5_1_1_2"/><b>then </b><i><font color="Green" title="E59">A97</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">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="c4">D1</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"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="E4:17_7_5_1_1_2"/><i><font color="Green" title="E60">A98</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E5:17_7_5_1_1_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_5_1_1"><i><font color="Green" title="E57">A95</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:17_7_5_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="c5">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E4:17_7_5_1_1_2"><i><font color="Green" title="E60">A98</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="E7:17_7_5_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></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="c4">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="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:17_7_5_1_1"><i><font color="Green" title="E57">A95</font></i></a>, <a class="txt" href="integra3.html#E3:17_7_5_1_1_2"><i><font color="Green" title="E59">A97</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="E4:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E5:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#E1:17_7_5_1_1"><i><font color="Green" title="E56">A94</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="E59">A99</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E12:17_7"><i><font color="Green" title="E55">A93</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E5:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="integra3.html#E11:17_7"><i><font color="Green" title="E54">A92</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_5_1_1"><i><font color="Green" title="E57">A95</font></i></a>, <a class="txt" href="integra3.html#E3:17_7_5_1_1"><i><font color="Green" title="E58">A96</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>
;<br/><a NAME="E7:17_7_5_1_1"/><i><font color="Green" title="E60">A100</font></i>: 
 <a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra3.html#T8" target="_self" title="INTEGRA3:th.8">Th8</a></i>;<br/><a NAME="E8:17_7_5_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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra3.html#E8:17"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E9:17_7_5_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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E7:17_7_5_1_1"><i><font color="Green" title="E60">A100</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="E10:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:17_7_5_1_1"><i><font color="Green" title="E59">A99</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_5_1_2"><b>suppose </b></a><a NAME="E1:17_7_5_1_2"/><i><font color="Green" title="E56">A101</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:17_7_5_1_2"/><i><font color="Green" title="E57">A102</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 1 <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/><a NAME="E3:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2 <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_1_2_1"/><i><font color="Green" title="E58">A103</font></i>: 
2 <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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D2</font>) <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="txt" href="integra3.html#E1:17_7_5_1_2"><i><font color="Green" title="E56">A101</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="c5">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#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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:17_7_5_1_2_1"><i><font color="Green" title="E58">A103</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="c5">D2</font>) <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> 2 <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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#E1:17_7_5_1_2"><i><font color="Green" title="E56">A101</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_5_1_2"><i><font color="Green" title="E57">A102</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:17_7_5_1_2"/><b>then </b><i><font color="Green" title="E58">A104</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E6:17_7_5_1_2"/><i><font color="Green" title="E59">A105</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="c4">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="c4">D1</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#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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_1_2_2"/><i><font color="Green" title="E60">A106</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">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c7">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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E7:17"><i><font color="Green" title="E23">A6</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="E2:17_7_5_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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c4">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="c4">D1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</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="E2:17_7_5_1_2_2"/><i><font color="Green" title="E61">A107</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="txt" href="integra3.html#E1:17_7_5_1_2"><i><font color="Green" title="E56">A101</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:17_7_5_1_2_2_2"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:17_7_5_1_2_2"/><i><font color="Green" title="E62">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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_1_2_2_3"/><i><font color="Green" title="E63">A109</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_5_1_2_2"><i><font color="Green" title="E61">A107</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">D2</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="E4:17_7_5_1_2_2_3"/><b>then </b><i><font color="Green" title="E64">A110</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">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="E5:17_7_5_1_2_2_3"/><i><font color="Green" title="E65">A111</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:17_7_5_1_2_2_3"/><i><font color="Green" title="E66">A112</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:17_7_5_1_2_2_3"/><i><font color="Green" title="E67">A113</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E8:17_7_5_1_2_2_3"/><i><font color="Green" title="E68">A114</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 
;<br/>
<b>thus </b><a NAME="E9:17_7_5_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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:17_7_5_1_2_2_3"><i><font color="Green" title="E63">A109</font></i></a>, <a class="txt" href="integra3.html#E6:17_7_5_1_2_2_3"><i><font color="Green" title="E66">A112</font></i></a>, <a class="txt" href="integra3.html#E7:17_7_5_1_2_2_3"><i><font color="Green" title="E67">A113</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="E10:17_7_5_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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:17_7_5_1_2_2_3"><i><font color="Green" title="E64">A110</font></i></a>, <a class="txt" href="integra3.html#E5:17_7_5_1_2_2_3"><i><font color="Green" title="E65">A111</font></i></a>, <a class="txt" href="integra3.html#E8:17_7_5_1_2_2_3"><i><font color="Green" title="E68">A114</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="E4:17_7_5_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="c4">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="c4">D1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17_7_5_1_2_2"><i><font color="Green" title="E60">A106</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_5_1_2_2"><i><font color="Green" title="E61">A107</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="c4">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="c4">D1</font></span>)</span></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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:17_7_5_1_2_2"><i><font color="Green" title="E62">A108</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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:17_7_5_1_2_2"><i><font color="Green" title="E62">A108</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:17_7_5_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="c4">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="c4">D1</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#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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E7:17_7_5_1_2"/><b>then </b><i><font color="Green" title="E60">A115</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span> <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> <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="c3">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="c4">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></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="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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/><a NAME="E8:17_7_5_1_2"/><i><font color="Green" title="E61">A116</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/><a NAME="E9:17_7_5_1_2"/><i><font color="Green" title="E62">A117</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/><a NAME="E10:17_7_5_1_2"/><i><font color="Green" title="E63">A118</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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></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="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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_1_2_3"/><i><font color="Green" title="E64">A119</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:17_7_5_1_2_3"/><b>then </b><i><font color="Green" title="E65">A120</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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">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="E3:17_7_5_1_2_3"/><i><font color="Green" title="E66">A121</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_5_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="txt" href="integra3.html#E1:17_7"><i><font color="Green" title="E48">A70</font></i></a>, <a class="txt" href="integra3.html#E2:17_7"><i><font color="Green" title="E49">A71</font></i></a>, <a class="txt" href="integra3.html#E1:17_7_5_1_2"><i><font color="Green" title="E56">A101</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:17_7_5_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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:17_7_5_1_2_3_1"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c8">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<b>set </b><font color="Maroon" title="c9">ID2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<i><font color="Green" title="E67">A122</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> = 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">D2</font>) <a href="goboard1.html#K1" title="GOBOARD1: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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E5:17_7_5_1_2"><i><font color="Green" title="E58">A104</font></i></a>, <a class="txt" href="integra3.html#E2:17_7_5_1_2_3"><i><font color="Green" title="E65">A120</font></i></a>, <a class="txt" href="integra3.html#E3:17_7_5_1_2_3"><i><font color="Green" title="E66">A121</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>
<br/>.= 
<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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E12:17_7"><i><font color="Green" title="E55">A93</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>
;<br/>
<a NAME="E5:17_7_5_1_2_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:17_7_5_1_2_3"><i><font color="Green" title="E64">A119</font></i></a>, <a class="txt" href="integra3.html#E3:17_7_5_1_2_3"><i><font color="Green" title="E66">A121</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<a NAME="E6:17_7_5_1_2_3"/><b>then </b><i><font color="Green" title="E68">A123</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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="c3">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:17"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/>
<a NAME="E7:17_7_5_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:17_7"><i><font color="Green" title="E55">A93</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="E8:17_7_5_1_2_3"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<a NAME="E9:17_7_5_1_2_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"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:17"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/>
<a NAME="E10:17_7_5_1_2_3"/><b>then </b><i><font color="Green" title="E69">A124</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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:17_7_5_1_2"><i><font color="Green" title="E61">A116</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:17_7_5_1_2_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#T10" title="INTEGRA1:th.10">INTEGRA1:10</a></i>;<br/>
<a NAME="E12:17_7_5_1_2_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"><font color="Maroon" title="c3">g</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="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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="c3">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:17"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="integra3.html#E13"><i><font color="Green" title="E13">Lm5</font></i></a></i>;<br/>
<a NAME="E13:17_7_5_1_2_3"/><b>then </b><i><font color="Green" title="E70">A125</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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c4">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> &amp; <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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c4">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E8:17_7_5_1_2"><i><font color="Green" title="E61">A116</font></i></a>, <a class="txt" href="integra3.html#E9:17_7_5_1_2"><i><font color="Green" title="E62">A117</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E14:17_7_5_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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E9:17_7_5_1_2"><i><font color="Green" title="E62">A117</font></i></a>, <a class="txt" href="integra3.html#E4:17_7_5_1_2_3"><i><font color="Green" title="E67">A122</font></i></a>, <a class="txt" href="integra3.html#E6:17_7_5_1_2_3"><i><font color="Green" title="E68">A123</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E15:17_7_5_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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E16:17_7_5_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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E17:17_7_5_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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E10:17_7_5_1_2_3"><i><font color="Green" title="E69">A124</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:17_7_5_1_2_3"/><b>then </b><i><font color="Green" title="E71">A126</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/>
<b>reconsider </b><font color="Maroon" title="c10">A</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</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="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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/>
<a NAME="E20:17_7_5_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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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="c4">D1</font>,<span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <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> <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="c3">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="c4">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E11:17_7"><i><font color="Green" title="E54">A92</font></i></a>, <a class="txt" href="integra3.html#E7:17_7_5_1_2"><i><font color="Green" title="E60">A115</font></i></a></i>;<br/>
<a NAME="E21:17_7_5_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="c4">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="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">g</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="c4">D1</font>,<span class="p0">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E13:17_7_5_1_2_3"><i><font color="Green" title="E70">A125</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E22:17_7_5_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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">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="c4">D1</font>,<span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E23:17_7_5_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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E13:17_7_5_1_2_3"><i><font color="Green" title="E70">A125</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:17_7_5_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="c4">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="c4">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">g</font></span>)</span></span>)</span> <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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E25:17_7_5_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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E18:17_7_5_1_2_3"><i><font color="Green" title="E71">A126</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="E26:17_7_5_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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></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="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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E11:17_7_5_1_2"/><i><font color="Green" title="E64">A127</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="c4">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="c4">D1</font></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="c4">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E14"><i><font color="Green" title="E14">Lm6</font></i></a></i>;<br/><a NAME="E12:17_7_5_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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra3.html#E8:17"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E13:17_7_5_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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <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="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E11:17_7_5_1_2"><i><font color="Green" title="E64">A127</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="E14:17_7_5_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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:17_7_5_1_2"><i><font color="Green" title="E59">A105</font></i></a>, <a class="txt" href="integra3.html#E10:17_7_5_1_2"><i><font color="Green" title="E63">A118</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="E14:17_7"/>
<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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></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"><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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E45:17"/><i><font color="Green" title="E48">A128</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8"><b>proof </b></a><div class="add">

<a NAME="E1:17_8"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">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="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="integra3.html#E6:17"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D21" title="INTEGRA1:def.21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:17_8"/><b>then </b><i><font color="Green" title="E49">A129</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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">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="E3:17_8"/><b>then </b><i><font color="Green" title="E50">A130</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="c5">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="E4:17_8"/><b>then </b><i><font color="Green" title="E51">A131</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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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">H<sub>2</sub></font>(<font color="Maroon" title="c5">D2</font>) )
 
<b>by </b><i><a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E5:17_8"/><i><font color="Green" title="E52">A132</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span> <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">D2</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:17_8"><i><font color="Green" title="E49">A129</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E6:17_8"/><i><font color="Green" title="E53">A133</font></i>: 
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E43:17"><i><font color="Green" title="E46">A68</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E7:17_8"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</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="E8:17_8"/><b>then </b>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">D2</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="E9:17_8"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</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">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E16:17"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="integra3.html#E4:17_8"><i><font color="Green" title="E51">A131</font></i></a>, <a class="txt" href="integra3.html#E6:17_8"><i><font color="Green" title="E53">A133</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">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E3:17_8"><i><font color="Green" title="E50">A130</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E11:17_8"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></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="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></span>)</span>)
 <b>by </b><i><a class="txt" href="integra3.html#E5:17_8"><i><font color="Green" title="E52">A132</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="E46:17"/><i><font color="Green" title="E49">A134</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_9"><b>proof </b></a><div class="add">

<a NAME="E1:17_9"/><i><font color="Green" title="E50">A135</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="c4">D1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font>) )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E2:17_9"/><i><font color="Green" title="E51">A136</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:17"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="integra1.html#D7" title="INTEGRA1:def.7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E3:17_9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E4:17_9"/><b>then </b><i><font color="Green" title="E52">A137</font></i>: 
<font color="Maroon" title="c7">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="c4">D1</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="E5:17_9"/>
<font color="Maroon" title="c7">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="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14:17"><i><font color="Green" title="E30">A16</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="E6:17_9"/><b>then </b>
<font color="Maroon" title="c7">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="c4">D1</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="E7:17_9"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">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="c4">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="real_1.html#K3" title="REAL_1:func.3">+</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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c7">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="c4">D1</font>),<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font>),1,<font color="Maroon" title="c7">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="c4">D1</font>),<span class="p3">(<span class="default"><font color="Maroon" title="c7">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</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="c4">D1</font>),1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E19:17"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="integra3.html#E1:17_9"><i><font color="Green" title="E50">A135</font></i></a>, <a class="txt" href="integra3.html#E4:17_9"><i><font color="Green" title="E52">A137</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="c4">D1</font>) <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:17_9"><i><font color="Green" title="E50">A135</font></i></a>, <a class="ref" href="jordan3.html#T25" title="JORDAN3:th.25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E9:17_9"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c7">j1</font>) <a href="real_1.html#K3" title="REAL_1:func.3">+</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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="c4">D1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font>)
 <b>by </b><i><a class="txt" href="integra3.html#E2:17_9"><i><font color="Green" title="E51">A136</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>
<i><font color="Green" title="E50">A138</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">D2</font>, <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">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="c4">D1</font></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="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">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="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E12:17"><i><font color="Green" title="E28">A14</font></i></a>, <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="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E11:17"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
;<br/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">D1</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13:17"><i><font color="Green" title="E29">A15</font></i></a>, <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="integra1.html#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13:17"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>
;<br/>
<b>hence </b><a NAME="E49:17"/>
<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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c4">D1</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#K14" title="INTEGRA1:func.14">upper_volume</a> <font color="Maroon" title="c3">g</font>,<font color="Maroon" title="c5">D2</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="c3">g</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</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="c3">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16" title="INTEGRA1:func.16">delta</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E42:17"><i><font color="Green" title="E45">A67</font></i></a>, <a class="txt" href="integra3.html#E44:17"><i><font color="Green" title="E47">A69</font></i></a>, <a class="txt" href="integra3.html#E45:17"><i><font color="Green" title="E48">A128</font></i></a>, <a class="txt" href="integra3.html#E46:17"><i><font color="Green" title="E49">A134</font></i></a>, <a class="txt" href="integra3.html#E47:17"><i><font color="Green" title="E50">A138</font></i></a></i>;<br/>


</div>
