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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of <font color="Maroon" title="c1">A</font>;<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 <font color="Maroon" title="c3">S</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:68"/><i><font color="Green" title="E49">A1</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>
 <b>and </b><br/><a NAME="E2:68"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<a NAME="E3:68"/>
for <font color="Olive" title="b1">i</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font> holds <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="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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="Olive" title="b1">i</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( $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> implies  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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>,$1</span>)</span></span>)</span> );<br/>
<a NAME="E1:68_1"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1_1"/><i><font color="Green" title="E52">A4</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font>
 ;<br/>

<a NAME="E2:68_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:68_1_1"/><b>then </b><i><font color="Green" title="E53">A6</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>
 
<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="E4:68_1_1"/><b>then </b><i><font color="Green" title="E54">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E5:68_1_1"/><b>then </b><i><font color="Green" title="E55">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E6:68_1_1"/><i><font color="Green" title="E56">A9</font></i>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1 <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> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_1"/><i><font color="Green" title="E57">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</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> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:68_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:68_1_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1: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"><font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_1_1_1"/><i><font color="Green" title="E58">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> )
 ;<br/>

<a NAME="E2:68_1_1_1_1"/><b>then </b><i><font color="Green" title="E59">A12</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E5:68_1_1"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:68_1_1_1_1"/><b>then </b><i><font color="Green" title="E60">A13</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:68_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E4:68_1_1_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_1_1"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:68_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</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="E6:68_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:68_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E8:68_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">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> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_1_1"><i><font color="Green" title="E59">A12</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_1_1_1"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1_1_1"/>
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1 <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> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_1"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:68_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>,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="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E8:68_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>,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="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:68_1_1"/><b>then </b><i><font color="Green" title="E57">A14</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>,1 &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>,1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="E10:68_1_1"/><b>then </b><i><font color="Green" title="E58">A15</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="c5">D2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:68_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" 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>,1</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>,1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E12:68_1_1"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" 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>,1</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>,1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E13:68_1_1"/><b>then </b><i><font color="Green" title="E60">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" 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>,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E14:68_1_1"/><i><font color="Green" title="E61">A18</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <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>,1</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1: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"><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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_1_2"/><i><font color="Green" title="E62">A19</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <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>,1</span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:68_1_1_2"/><b>then </b><i><font color="Green" title="E63">A20</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">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>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E13:68_1_1"><i><font color="Green" title="E60">A17</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:68_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="integra1.html#E10:68_1_1"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_2"><i><font color="Green" title="E62">A19</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E4:68_1_1_2"/><b>then </b><i><font color="Green" title="E64">A21</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E5:68_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_2"><i><font color="Green" title="E63">A20</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="E6:68_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:68_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E8:68_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_2"><i><font color="Green" title="E64">A21</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:68_1_1"/><b>then </b><i><font color="Green" title="E62">A22</font></i>: 
 <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>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E13:68_1_1"><i><font color="Green" title="E60">A17</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">DD1</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1;<br/>
<b>set </b><font color="Maroon" title="c7">DD2</font> =  <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>,1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">B</font> =  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1;<br/>
<b>consider </b><font color="Maroon" title="c9">S1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1;<br/>
<a NAME="E17:68_1_1"/><i><font color="Green" title="E63">A23</font></i>: 
<font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="c4">D1</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<a NAME="E18:68_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>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">DD1</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,1,1 as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c9">S1</font> <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E17:68_1_1"><i><font color="Green" title="E63">A23</font></i></a>, <a class="ref" href="integra1.html#T39" target="_self" title="INTEGRA1:th.39">Th39</a></i>;<br/>
<a NAME="E20:68_1_1"/><i><font color="Green" title="E64">A24</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>,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="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E21:68_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>,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="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:68_1_1"/><b>then </b><i><font color="Green" title="E65">A25</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>,1 &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>,1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">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="E23:68_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:68_1_1"/><b>then </b>
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="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E25:68_1_1"/><b>then </b><i><font color="Green" title="E66">A26</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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="E26:68_1_1"/><i><font color="Green" title="E67">A27</font></i>: 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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>,1</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="c4">D1</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E17:68_1_1"><i><font color="Green" title="E63">A23</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E27:68_1_1"/><i><font color="Green" title="E68">A28</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></i>;<br/>
<a NAME="E28:68_1_1"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E25:68_1_1"><i><font color="Green" title="E66">A26</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="E29:68_1_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E27:68_1_1"><i><font color="Green" title="E68">A28</font></i></a></i>;<br/>
<a NAME="E30:68_1_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font></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" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E31:68_1_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E32:68_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
;<br/>
<a NAME="E33:68_1_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">DD2</font> =  <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>,1</span>)</span> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c9">S1</font> <b>by </b><i><a class="txt" href="integra1.html#E20:68_1_1"><i><font color="Green" title="E64">A24</font></i></a>, <a class="txt" href="integra1.html#E22:68_1_1"><i><font color="Green" title="E65">A25</font></i></a>, <a class="txt" href="integra1.html#E25:68_1_1"><i><font color="Green" title="E66">A26</font></i></a>, <a class="txt" href="integra1.html#E26:68_1_1"><i><font color="Green" title="E67">A27</font></i></a>, <a class="ref" href="integra1.html#T39" target="_self" title="INTEGRA1:th.39">Th39</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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>,1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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>,1</span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T43" title="PARTFUN1:th.43">PARTFUN1:43</a></i>;<br/>
<a NAME="E36:68_1_1"/><i><font color="Green" title="E69">A29</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E37:68_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E38:68_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</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="c4">D1</font>,1
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self" title="INTEGRA1:th.10">Th10</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E39:68_1_1"/><b>then </b>
<font color="Maroon" title="c13">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E40:68_1_1"/><b>then </b><i><font color="Green" title="E70">A30</font></i>: 
<font color="Maroon" title="c13">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a></i>;<br/>
<a NAME="E41:68_1_1"/>
<font color="Maroon" title="c13">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c14">a</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E2:68_1_1_3"/><i><font color="Green" title="E71">A31</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Maroon" title="c14">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E3:68_1_1_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font></span>)</span> holds <br/><font color="Maroon" title="c14">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:68_1_1_3_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font></span>)</span>
 ;<br/>

<a NAME="E2:68_1_1_3_1"/><b>then </b><i><font color="Green" title="E72">A32</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:68_1_1_3_1"/><i><font color="Green" title="E73">A33</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:68_1_1_3_1"/><i><font color="Green" title="E74">A34</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">x</font> = <font color="Maroon" title="c15">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_3_1"><i><font color="Green" title="E72">A32</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_1_3_1"><i><font color="Green" title="E73">A33</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:68_1_1_3_1"/>
<font color="Maroon" title="c14">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">x</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_3"><i><font color="Green" title="E71">A31</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_3_1"><i><font color="Green" title="E72">A32</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_1_3_1"><i><font color="Green" title="E73">A33</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_1_3_1"><i><font color="Green" title="E74">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:68_1_1_3_1"/>
<font color="Maroon" title="c14">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_3_1"><i><font color="Green" title="E72">A32</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_1_1_3"/>
<font color="Maroon" title="c13">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E42:68_1_1"/><b>then </b><i><font color="Green" title="E71">A35</font></i>: 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="integra1.html#E40:68_1_1"><i><font color="Green" title="E70">A30</font></i></a>, <a class="ref" href="integra1.html#T33" target="_self" title="INTEGRA1:th.33">Th33</a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E43:68_1_1"/><i><font color="Green" title="E72">A36</font></i>: 
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_4"><b>proof </b></a><div class="add">

<i><font color="Green" title="E73">A37</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">DD1</font>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>
;<br/>
<a NAME="E2:68_1_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:68_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/>
<a NAME="E3:68_1_1_4"/><b>then </b><i><font color="Green" title="E74">A38</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_4"><i><font color="Green" title="E73">A37</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:68_1_1_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_1_4_2"/><i><font color="Green" title="E75">A39</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> )
 ;<br/>

<a NAME="E2:68_1_1_4_2"/><i><font color="Green" title="E76">A40</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">DD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E3:68_1_1_4_2"/><i><font color="Green" title="E77">A41</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E4:68_1_1_4_2"/><i><font color="Green" title="E78">A43</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">DD1</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_4_2"><i><font color="Green" title="E76">A40</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_1_4_2"><i><font color="Green" title="E77">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:68_1_1_4_2"/><i><font color="Green" title="E79">A44</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_4_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E2:68_1_1_4_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_1_4_2"><i><font color="Green" title="E77">A41</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E80">A45</font></i>: <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_4"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_4_2"><i><font color="Green" title="E75">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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="c10">DD1</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c10">DD1</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_4_2"><i><font color="Green" title="E78">A43</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>
;<br/>
<i><font color="Green" title="E81">A46</font></i>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c10">DD1</font>,1 = 
<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="c10">DD1</font>,1</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="c10">DD1</font>,1</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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>,1</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="c10">DD1</font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_4_2"><i><font color="Green" title="E78">A43</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">DD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_4_2"><i><font color="Green" title="E78">A43</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_1"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E5:68_1_1_4_2"><i><font color="Green" title="E79">A44</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<i><font color="Green" title="E82">A47</font></i>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1 = 
<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>,1</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>,1</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></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>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>
;<br/>
<b>then </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E3:68_1_1"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></i>
;<br/>
<a NAME="E11:68_1_1_4_2"/><b>then </b><i><font color="Green" title="E83">A48</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></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/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_4"><i><font color="Green" title="E73">A37</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_4_2"><i><font color="Green" title="E75">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E11:68_1_1_4_2"><i><font color="Green" title="E83">A48</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:68_1_1_4_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E36:68_1_1"><i><font color="Green" title="E69">A29</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_1_4_2"><i><font color="Green" title="E80">A45</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_1_4_2"><i><font color="Green" title="E81">A46</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_1_4_2"><i><font color="Green" title="E82">A47</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:68_1_1_4"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c10">DD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E3:68_1_1_4"><i><font color="Green" title="E74">A38</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E44:68_1_1"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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>,1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5"/><i><font color="Green" title="E73">A49</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</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>,1
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/>
<a NAME="E2:68_1_1_5"/>
 <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/>
<a NAME="E3:68_1_1_5"/><b>then </b><i><font color="Green" title="E74">A50</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5"><i><font color="Green" title="E73">A49</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:68_1_1_5"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_1_5_1"/><i><font color="Green" title="E75">A51</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font></span>)</span> )
 ;<br/>

<a NAME="E2:68_1_1_5_1"/><i><font color="Green" title="E76">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">DD2</font> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">DD2</font></span>)</span> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">DD2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_1_1_5_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">DD2</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1"><i><font color="Green" title="E75">A51</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/>

<b>hence </b><a NAME="E2:68_1_1_5_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">DD2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>

<b>hence </b><a NAME="E3:68_1_1_5_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">DD2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:68_1_1_5_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c14">i</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>,<font color="Maroon" title="c14">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_2"/><i><font color="Green" title="E77">A53</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">DD2</font> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_1_1_5_1_2_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">DD2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a></i>;<br/>

<a NAME="E2:68_1_1_5_1_2_1"/>
 <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>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:68_1_1_5_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:68_1_1_5_1_2_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_5_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_5_1_2_2_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_5_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_5_1_2_2_1_1"/><i><font color="Green" title="E78">A54</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_1_5_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E79">A55</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E15:68_1_1"><i><font color="Green" title="E62">A22</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:68_1_1_5_1_2_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E4:68_1_1_5_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E80">A56</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><i><font color="Green" title="E81">A57</font></i>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c14">i</font> = 
<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="c11">DD2</font>,1</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="c11">DD2</font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E78">A54</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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>,1</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="c11">DD2</font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E78">A54</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E78">A54</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_1"><i><font color="Green" title="E61">A18</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E78">A54</font></i></a></i>
<br/>.= 
<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>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<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>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E79">A55</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/> <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E78">A54</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E80">A56</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E80">A56</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/><b>hence </b><a NAME="E7:68_1_1_5_1_2_2_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c14">i</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>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E5:68_1_1_5_1_2_2_1_1"><i><font color="Green" title="E81">A57</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_5_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_5_1_2_2_1_2"/><i><font color="Green" title="E78">A58</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_1_5_1_2_2_1_2"/><i><font color="Green" title="E79">A59</font></i>: 
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_2_2_1_2_1"/><i><font color="Green" title="E80">A60</font></i>: 
<font color="Maroon" title="c14">i</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c15">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:68_1_1_5_1_2_2_1_2_1_1"/><i><font color="Green" title="E81">A61</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E3:68_1_1_5_1_2_2_1_2_1_1"/>
not <font color="Maroon" title="c14">i</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<a NAME="E4:68_1_1_5_1_2_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E82">A62</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E5:68_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,1 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E6:68_1_1_5_1_2_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E83">A63</font></i>: 
<font color="Maroon" title="c14">i</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#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>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1_2_2_1_2_1_1"><i><font color="Green" title="E81">A61</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:68_1_1_5_1_2_2_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="integra1.html#E22:68_1_1"><i><font color="Green" title="E65">A25</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_1_5_1_2_2_1_2_1_1"><i><font color="Green" title="E82">A62</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:68_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">i</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#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="integra1.html#E2:68_1_1_5_1_2_2_1_2_1_1"><i><font color="Green" title="E81">A61</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:68_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">i</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="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="E10:68_1_1_5_1_2_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">i</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> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E6:68_1_1_5_1_2_2_1_2_1_1"><i><font color="Green" title="E83">A63</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:68_1_1_5_1_2_2_1_2_1_1"/>
<font color="Maroon" title="c14">i</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> = 
<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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E13:68_1_1"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="integra1.html#E14:68_1_1"><i><font color="Green" title="E61">A18</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
;<br/>
<b>hence </b><a NAME="E3:68_1_1_5_1_2_2_1_2_1"/>
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2_1"><i><font color="Green" title="E80">A60</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:68_1_1_5_1_2_2_1_2"/><i><font color="Green" title="E80">A64</font></i>: 
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2_2_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_2_2_1_2_2"/><i><font color="Green" title="E81">A65</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_2_2_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_2_2_1_2_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E22:68_1_1"><i><font color="Green" title="E65">A25</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:68_1_1_5_1_2_2_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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="E3:68_1_1_5_1_2_2_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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="E4:68_1_1_5_1_2_2_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:68_1_1_5_1_2_2_1_2_2_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="integra1.html#E13:68_1_1"><i><font color="Green" title="E60">A17</font></i></a>, <a class="txt" href="integra1.html#E14:68_1_1"><i><font color="Green" title="E61">A18</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
;<br/>
<b>hence </b><a NAME="E3:68_1_1_5_1_2_2_1_2_2"/>
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2_2"><i><font color="Green" title="E81">A65</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E81">A66</font></i>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c14">i</font> = 
<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="c11">DD2</font>,<font color="Maroon" title="c14">i</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="c11">DD2</font>,<font color="Maroon" title="c14">i</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</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="c11">DD2</font>,<font color="Maroon" title="c14">i</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="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E79">A59</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E80">A64</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/> <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</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="c5">D2</font>,<font color="Maroon" title="c14">i</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</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="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_2"><i><font color="Green" title="E77">A53</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E78">A58</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/><b>hence </b><a NAME="E6:68_1_1_5_1_2_2_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c14">i</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>,<font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_5_1_2_2_1_2"><i><font color="Green" title="E81">A66</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1_1_5_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c14">i</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>,<font color="Maroon" title="c14">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:68_1_1_5_1"/><b>then </b><i><font color="Green" title="E77">A67</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">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>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/>
<a NAME="E5:68_1_1_5_1"/><i><font color="Green" title="E78">A68</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_3"/><i><font color="Green" title="E79">A69</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</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="c5">D2</font>,<font color="Maroon" title="c14">i</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E2:68_1_1_5_1_3"/><i><font color="Green" title="E80">A70</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1 <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>,1</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>,1</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E3:68_1_1_5_1_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>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_3_1"/><i><font color="Green" title="E81">A71</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">DD2</font> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:68_1_1_5_1_3_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">DD2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a></i>;<br/>

<a NAME="E2:68_1_1_5_1_3_1_1"/>
 <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>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:68_1_1_5_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:68_1_1_5_1_3_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_5_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_5_1_3_1_2_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_5_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_5_1_3_1_2_1_1"/><i><font color="Green" title="E82">A72</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_1_5_1_3_1_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E15:68_1_1"><i><font color="Green" title="E62">A22</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1"><i><font color="Green" title="E81">A71</font></i></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/><a NAME="E3:68_1_1_5_1_3_1_2_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E4:68_1_1_5_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E83">A73</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:68_1_1_5_1_3_1_2_1_1"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E82">A72</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E6:68_1_1_5_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E84">A74</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>,<font color="Maroon" title="c14">i</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="c4">D1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E7:68_1_1_5_1_3_1_2_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>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_3_1_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_3_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <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="c4">D1</font>,1</span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:68_1_1_5_1_3_1_2_1_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:68_1_1_5_1_3_1_2_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra1.html#E6:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E84">A74</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/><a NAME="E9:68_1_1_5_1_3_1_2_1_1"/><i><font color="Green" title="E85">A75</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E82">A72</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E83">A73</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E10:68_1_1_5_1_3_1_2_1_1"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E20:68_1_1"><i><font color="Green" title="E64">A24</font></i></a>, <a class="txt" href="integra1.html#E22:68_1_1"><i><font color="Green" title="E65">A25</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E83">A73</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E11:68_1_1_5_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/><a NAME="E12:68_1_1_5_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E86">A76</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E9:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E85">A75</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E13:68_1_1_5_1_3_1_2_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_3_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_3_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b1">a</font> <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>,<font color="Maroon" title="c14">i</font></span>)</span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E2:68_1_1_5_1_3_1_2_1_1_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c15">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E15:68_1_1_5_1_3_1_2_1_1"/><i><font color="Green" title="E87">A77</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Maroon" title="c15">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E6:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E84">A74</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E86">A76</font></i></a></i>;<br/><a NAME="E16:68_1_1_5_1_3_1_2_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra1.html#E15:68_1_1_5_1_3_1_2_1_1"><i><font color="Green" title="E87">A77</font></i></a></i>;<br/><b>hence </b><a NAME="E17:68_1_1_5_1_3_1_2_1_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</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="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_5_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E82">A78</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c15">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E83">A79</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E4:68_1_1_5_1_3_1_2_1_2"/>
not <font color="Maroon" title="c14">i</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E82">A78</font></i></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/><a NAME="E5:68_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E84">A80</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E6:68_1_1_5_1_3_1_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="integra1.html#E22:68_1_1"><i><font color="Green" title="E65">A25</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:68_1_1_5_1_3_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">i</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#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="integra1.html#E3:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E83">A79</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:68_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E85">A81</font></i>: 
<font color="Maroon" title="c14">i</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="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="E9:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E86">A82</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">A</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a> )
 <b>by </b><i><a class="ref" href="integra1.html#T3" target="_self" title="INTEGRA1:th.3">Th3</a></i>;<br/><a NAME="E10:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E87">A83</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>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1"><i><font color="Green" title="E81">A71</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E82">A78</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E11:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E88">A84</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1"><i><font color="Green" title="E81">A71</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E82">A78</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E12:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E89">A85</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>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E13:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E90">A86</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/><a NAME="E14:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E91">A87</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></i>;<br/><a NAME="E15:68_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</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>  <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="integra1.html#E8:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E85">A81</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="E16:68_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E92">A88</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>,<font color="Maroon" title="c14">i</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E86">A82</font></i></a>, <a class="txt" href="integra1.html#E10:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E87">A83</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E89">A85</font></i></a>, <a class="txt" href="integra1.html#E14:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E91">A87</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><a NAME="E17:68_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E20:68_1_1"><i><font color="Green" title="E64">A24</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E84">A80</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E85">A81</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E18:68_1_1_5_1_3_1_2_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>,<font color="Maroon" title="c14">i</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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E10:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E87">A83</font></i></a>, <a class="txt" href="integra1.html#E13:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E90">A86</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E20:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E93">A89</font></i>: 
( <font color="Maroon" title="c16">a</font> <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>,<font color="Maroon" title="c14">i</font></span>)</span> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">a</font> &amp; <font color="Maroon" title="c16">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E16:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E92">A88</font></i></a></i>;<br/><a NAME="E21:68_1_1_5_1_3_1_2_1_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra1.html#E20:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E93">A89</font></i></a></i>;<br/><b>hence </b><a NAME="E22:68_1_1_5_1_3_1_2_1_2"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</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="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E23:68_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c5">D2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1"><i><font color="Green" title="E81">A71</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="E24:68_1_1_5_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E94">A90</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</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="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E9:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E86">A82</font></i></a>, <a class="txt" href="integra1.html#E11:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E88">A84</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E89">A85</font></i></a>, <a class="txt" href="integra1.html#E14:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E91">A87</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><a NAME="E25:68_1_1_5_1_3_1_2_1_2"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E20:68_1_1"><i><font color="Green" title="E64">A24</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1_5_1_3_1"><i><font color="Green" title="E81">A71</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E26:68_1_1_5_1_3_1_2_1_2"/><b>then </b>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="integra1.html#E11:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E88">A84</font></i></a>, <a class="txt" href="integra1.html#E13:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E90">A86</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E28:68_1_1_5_1_3_1_2_1_2"/><i><font color="Green" title="E95">A91</font></i>: 
( <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">a</font> &amp; <font color="Maroon" title="c17">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E24:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E94">A90</font></i></a></i>;<br/><a NAME="E29:68_1_1_5_1_3_1_2_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c4">D1</font>,1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra1.html#E28:68_1_1_5_1_3_1_2_1_2"><i><font color="Green" title="E95">A91</font></i></a></i>;<br/><b>hence </b><a NAME="E30:68_1_1_5_1_3_1_2_1_2"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</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="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1_1_5_1_3_1"/>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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>,<font color="Maroon" title="c14">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,1</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>,1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_1_1_5_1_3"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,1
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_3"><i><font color="Green" title="E79">A69</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1_3"><i><font color="Green" title="E80">A70</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:68_1_1_5_1"/><i><font color="Green" title="E79">A92</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font> &amp; <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1_5_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_1_5_1_4"/><i><font color="Green" title="E80">A93</font></i>: 
 <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>,1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="integra1.html#E9:68_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E2:68_1_1_5_1_4"/><b>then </b>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:68_1_1_5_1_4"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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 href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<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></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>
<br/>.= 
 <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>,1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_1_5_1_4"><i><font color="Green" title="E80">A93</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E5:68_1_1_5_1_4"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E12:68_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_1_5_1"><i><font color="Green" title="E76">A52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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>,1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<b>by </b><i><a class="txt" href="integra1.html#E6:68_1_1_5_1"><i><font color="Green" title="E79">A92</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c14">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E6:68_1_1_5_1"><i><font color="Green" title="E79">A92</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>
;<br/>
<b>hence </b><a NAME="E8:68_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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>,1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E4:68_1_1_5_1"><i><font color="Green" title="E77">A67</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_1_5_1"><i><font color="Green" title="E78">A68</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:68_1_1_5"/>
 <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c11">DD2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E3:68_1_1_5"><i><font color="Green" title="E74">A50</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E45:68_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E42:68_1_1"><i><font color="Green" title="E71">A35</font></i></a>, <a class="txt" href="integra1.html#E43:68_1_1"><i><font color="Green" title="E72">A36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:68_1"/><i><font color="Green" title="E52">A94</font></i>: 
for <font color="Olive" title="b1">k</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:68_1_2"/><i><font color="Green" title="E53">A95</font></i>: 
( <font color="Maroon" title="c6">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> implies  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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="c6">k</font></span>)</span></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:68_1_2"/><i><font color="Green" title="E54">A96</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font>
 ;<br/>

<a NAME="E3:68_1_2"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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/>
<a NAME="E4:68_1_2"/><b>then </b><i><font color="Green" title="E55">A98</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_2_1_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:68_1_2_1_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_2_1_1_2"/><i><font color="Green" title="E56">A99</font></i>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E2:68_1_2_1_1_2"/>
not <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/><a NAME="E3:68_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <b>by </b><i><a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a></i>;<br/><a NAME="E4:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E57">A100</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <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="E5:68_1_2_1_1_2"/><i><font color="Green" title="E58">A101</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E6:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E59">A102</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">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="integra1.html#E4:68_1_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2"><i><font color="Green" title="E57">A100</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A103</font></i>: 
<font color="Maroon" title="c6">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="E8:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E61">A104</font></i>: 
<font color="Maroon" title="c6">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="E9:68_1_2_1_1_2"/><i><font color="Green" title="E62">A105</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/><a NAME="E10:68_1_2_1_1_2"/><i><font color="Green" title="E63">A106</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font>
 <b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/><a NAME="E11:68_1_2_1_1_2"/><i><font color="Green" title="E64">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>,<font color="Maroon" title="c6">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="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/><a NAME="E12:68_1_2_1_1_2"/><i><font color="Green" title="E65">A108</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/><a NAME="E13:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E66">A109</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#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="E14:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E67">A110</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="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="E15:68_1_2_1_1_2"/><i><font color="Green" title="E68">A111</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E10:68_1_2_1_1_2"><i><font color="Green" title="E63">A106</font></i></a>, <a class="txt" href="integra1.html#E13:68_1_2_1_1_2"><i><font color="Green" title="E66">A109</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E16:68_1_2_1_1_2"/><i><font color="Green" title="E69">A112</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="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1_2_1_1_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="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E70">A113</font></i>: 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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="c6">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:68_1_2_1_1_2"><i><font color="Green" title="E64">A107</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_2_1_1_2"><i><font color="Green" title="E65">A108</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E71">A114</font></i>: 
<font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E5:68_1_2_1_1_2_1"/>
<font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<b>hence </b><a NAME="E6:68_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_1"><i><font color="Green" title="E70">A113</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_1"><i><font color="Green" title="E71">A114</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:68_1_2_1_1_2"/><i><font color="Green" title="E70">A115</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1_2_1_1_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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c6">k</font>
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E71">A116</font></i>: 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:68_1_2_1_1_2"><i><font color="Green" title="E64">A107</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_2_1_1_2"><i><font color="Green" title="E65">A108</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_2"/>
<font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_2"><i><font color="Green" title="E71">A116</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<b>hence </b><a NAME="E5:68_1_2_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c8">ID</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:68_1_2_1_1_2"/><i><font color="Green" title="E71">A117</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">ID</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">ID</font> = <font color="Maroon" title="c8">ID</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E21:68_1_2_1_1_2"/><i><font color="Green" title="E72">A118</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E6:68_1_2_1_1_2"><i><font color="Green" title="E59">A102</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/><a NAME="E22:68_1_2_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="c6">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="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/><a NAME="E23:68_1_2_1_1_2"/><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="c6">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="E24:68_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E73">A119</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="c6">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="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E10:68_1_2_1_1_2"><i><font color="Green" title="E63">A106</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>set </b><font color="Maroon" title="c10">K1D1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">KD1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font>;<br/><b>set </b><font color="Maroon" title="c12">K1D2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c13">KD2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="c6">k</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">IDK1</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c15">IDK</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="c6">k</font>;<br/><a NAME="E25:68_1_2_1_1_2"/><i><font color="Green" title="E74">A120</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E9:68_1_2_1_1_2"><i><font color="Green" title="E62">A105</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">p1</font>, <font color="Maroon" title="c17">q1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E27:68_1_2_1_1_2"/><i><font color="Green" title="E75">A121</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">q1</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/><a NAME="E28:68_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="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="c6">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E15:68_1_2_1_1_2"><i><font color="Green" title="E68">A111</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">p2</font>, <font color="Maroon" title="c19">q2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E30:68_1_2_1_1_2"/><i><font color="Green" title="E76">A122</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</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="c6">k</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c19">q2</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E19:68_1_2_1_1_2"><i><font color="Green" title="E71">A117</font></i></a>, <a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/><a NAME="E31:68_1_2_1_1_2"/><i><font color="Green" title="E77">A123</font></i>: 
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_3"/><i><font color="Green" title="E78">A124</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E21:68_1_2_1_1_2"><i><font color="Green" title="E72">A118</font></i></a>, <a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:68_1_2_1_1_2_3"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p1</font> holds <br/><font color="Maroon" title="c16">p1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_2_1_1_2_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p1</font> )
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E79">A125</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p1</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="E3:68_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E80">A126</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">p1</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="E4:68_1_2_1_1_2_3_1"/><i><font color="Green" title="E81">A127</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_3"><i><font color="Green" title="E78">A124</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_3_1"><i><font color="Green" title="E79">A125</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:68_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E82">A128</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<i><font color="Green" title="E83">A129</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="integra1.html#E5:68_1_2_1_1_2_3_1"><i><font color="Green" title="E82">A128</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<i><font color="Green" title="E84">A130</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="integra1.html#E21:68_1_2_1_1_2"><i><font color="Green" title="E72">A118</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E8:68_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E9:68_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E85">A131</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c6">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E9:68_1_2_1_1_2"><i><font color="Green" title="E62">A105</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E11:68_1_2_1_1_2_3_1"/><b>then </b>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2_1_1_2_3_1"><i><font color="Green" title="E81">A127</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_2_1_1_2_3_1"><i><font color="Green" title="E84">A130</font></i></a>, <a class="txt" href="integra1.html#E9:68_1_2_1_1_2_3_1"><i><font color="Green" title="E85">A131</font></i></a></i>;<br/>
<a NAME="E12:68_1_2_1_1_2_3_1"/><b>then </b><i><font color="Green" title="E86">A132</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_2_1_1_2_3_1"><i><font color="Green" title="E86">A132</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E14:68_1_2_1_1_2_3_1"/>
<font color="Maroon" title="c16">p1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_3_1"><i><font color="Green" title="E80">A126</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_3_1"><i><font color="Green" title="E83">A129</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1_2_1_1_2_3"/>
<font color="Maroon" title="c16">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_3"><i><font color="Green" title="E78">A124</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E32:68_1_2_1_1_2"/><i><font color="Green" title="E78">A133</font></i>: 
<font color="Maroon" title="c18">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="c6">k</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_4"/><i><font color="Green" title="E79">A134</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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="c6">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E24:68_1_2_1_1_2"><i><font color="Green" title="E73">A119</font></i></a>, <a class="txt" href="integra1.html#E30:68_1_2_1_1_2"><i><font color="Green" title="E76">A122</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E2:68_1_2_1_1_2_4"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</font> holds <br/><font color="Maroon" title="c18">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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="c6">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_2_1_1_2_4_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</font> )
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E80">A135</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">p2</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="E3:68_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E81">A136</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">p2</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="E4:68_1_2_1_1_2_4_1"/><i><font color="Green" title="E82">A137</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="c6">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_4"><i><font color="Green" title="E79">A134</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_4_1"><i><font color="Green" title="E80">A135</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:68_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E83">A138</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c6">k</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<i><font color="Green" title="E84">A139</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="c6">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c6">k</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="integra1.html#E5:68_1_2_1_1_2_4_1"><i><font color="Green" title="E83">A138</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<i><font color="Green" title="E85">A140</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="c6">k</font></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <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="c6">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E24:68_1_2_1_1_2"><i><font color="Green" title="E73">A119</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E8:68_1_2_1_1_2_4_1"/><i><font color="Green" title="E86">A141</font></i>: 
 <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="c6">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E17:68_1_2_1_1_2"><i><font color="Green" title="E70">A115</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <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>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E15:68_1_2_1_1_2"><i><font color="Green" title="E68">A111</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E10:68_1_2_1_1_2_4_1"/><b>then </b>
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2_1_1_2_4_1"><i><font color="Green" title="E82">A137</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_2_1_1_2_4_1"><i><font color="Green" title="E85">A140</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2_4_1"><i><font color="Green" title="E86">A141</font></i></a></i>;<br/>
<a NAME="E11:68_1_2_1_1_2_4_1"/><b>then </b><i><font color="Green" title="E87">A142</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="integra1.html#E11:68_1_2_1_1_2_4_1"><i><font color="Green" title="E87">A142</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b><a NAME="E13:68_1_2_1_1_2_4_1"/>
<font color="Maroon" title="c18">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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="c6">k</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="integra1.html#E30:68_1_2_1_1_2"><i><font color="Green" title="E76">A122</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_4_1"><i><font color="Green" title="E81">A136</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_4_1"><i><font color="Green" title="E84">A139</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68_1_2_1_1_2_4"/>
<font color="Maroon" title="c18">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_4"><i><font color="Green" title="E79">A134</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E33:68_1_2_1_1_2"/><i><font color="Green" title="E79">A143</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">q2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c20">DD1</font> =  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c21">MD1</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c22">MD2</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c23">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c24">S1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="integra1.html#M2" title="INTEGRA1:mode.2">Division</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E2:68_1_2_1_1_2_5"/><i><font color="Green" title="E80">A144</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 
;<br/>
<a NAME="E2:68_1_2_1_1_2_5_1"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:68_1_2_1_1_2_5_1"/>
( <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2"><i><font color="Green" title="E58">A101</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c25">MD1</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c24">S1</font> <b>by </b><i><a class="ref" href="integra1.html#T39" target="_self" title="INTEGRA1:th.39">Th39</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_5"/><i><font color="Green" title="E81">A145</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E16:68_1_2_1_1_2"><i><font color="Green" title="E69">A112</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:68_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E82">A146</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:68_1_2_1_1_2_5"/><i><font color="Green" title="E83">A147</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E7:68_1_2_1_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5"><i><font color="Green" title="E82">A146</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:68_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E84">A148</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="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="E9:68_1_2_1_1_2_5"/><i><font color="Green" title="E85">A149</font></i>: 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<a NAME="E10:68_1_2_1_1_2_5"/><i><font color="Green" title="E86">A150</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<a NAME="E11:68_1_2_1_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 
;<br/>
<a NAME="E12:68_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E87">A151</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<a NAME="E13:68_1_2_1_1_2_5"/>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1: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="c6">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E11:68_1_2_1_1_2"><i><font color="Green" title="E64">A107</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5"><i><font color="Green" title="E84">A148</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E14:68_1_2_1_1_2_5"/><b>then </b>
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_2_1_1_2_5"><i><font color="Green" title="E87">A151</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">MD2</font> =  <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of <font color="Maroon" title="c24">S1</font> <b>by </b><i><a class="txt" href="integra1.html#E12:68_1_2_1_1_2"><i><font color="Green" title="E65">A108</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5"><i><font color="Green" title="E81">A145</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5"><i><font color="Green" title="E84">A148</font></i></a>, <a class="txt" href="integra1.html#E9:68_1_2_1_1_2_5"><i><font color="Green" title="E85">A149</font></i></a>, <a class="txt" href="integra1.html#E10:68_1_2_1_1_2_5"><i><font color="Green" title="E86">A150</font></i></a>, <a class="ref" href="integra1.html#T39" target="_self" title="INTEGRA1:th.39">Th39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">g</font> = <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T43" title="PARTFUN1:th.43">PARTFUN1:43</a></i>;<br/>
<a NAME="E17:68_1_2_1_1_2_5"/><i><font color="Green" title="E88">A152</font></i>: 
( <font color="Maroon" title="c27">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c27">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_5_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E2:68_1_2_1_1_2_5_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_5_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</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="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5"><i><font color="Green" title="E80">A144</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self" title="INTEGRA1:th.10">Th10</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E4:68_1_2_1_1_2_5_2"/>
<font color="Maroon" title="c27">g</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E5:68_1_2_1_1_2_5_2"/>
<font color="Maroon" title="c27">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c28">a</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E2:68_1_2_1_1_2_5_2_1"/><i><font color="Green" title="E89">A153</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Maroon" title="c28">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_5_2_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font></span>)</span> holds <br/><font color="Maroon" title="c28">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c29">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:68_1_2_1_1_2_5_2_1_1"/>
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font></span>)</span>
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_5_2_1_1"/><b>then </b><i><font color="Green" title="E90">A154</font></i>: 
<font color="Maroon" title="c29">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_5_2_1_1"/><i><font color="Green" title="E91">A155</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_5_2_1_1"/><i><font color="Green" title="E92">A156</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c27">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">x</font> = <font color="Maroon" title="c29">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_2_1_1"><i><font color="Green" title="E90">A154</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_2_1_1"><i><font color="Green" title="E91">A155</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:68_1_2_1_1_2_5_2_1_1"/>
<font color="Maroon" title="c28">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_2_1"><i><font color="Green" title="E89">A153</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_2_1_1"><i><font color="Green" title="E90">A154</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_2_1_1"><i><font color="Green" title="E91">A155</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_2_1_1"><i><font color="Green" title="E92">A156</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:68_1_2_1_1_2_5_2_1_1"/>
<font color="Maroon" title="c28">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_2_1_1"><i><font color="Green" title="E90">A154</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_1_2_1_1_2_5_2_1"/>
<font color="Maroon" title="c27">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10" title="RFUNCT_1:def.10">RFUNCT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:68_1_2_1_1_2_5_2"/>
<font color="Maroon" title="c27">g</font> <a href="rfunct_1.html#R2" title="RFUNCT_1:pred.2">is_bounded_below_on</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:68_1_2_1_1_2_5"/><b>then </b><i><font color="Green" title="E89">A157</font></i>: 
<font color="Maroon" title="c27">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T131" title="FUNCT_2:th.131">FUNCT_2:131</a></i>;<br/>
<a NAME="E19:68_1_2_1_1_2_5"/><i><font color="Green" title="E90">A158</font></i>: 
<font color="Maroon" title="c17">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_3"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_5_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:68_1_2_1_1_2_5_3"/><b>then </b><i><font color="Green" title="E91">A159</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_5_3"/><b>then </b><i><font color="Green" title="E92">A160</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_5_3"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">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="c17">q1</font> holds <br/><font color="Maroon" title="c17">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_2_1_1_2_5_3_1"/><i><font color="Green" title="E93">A161</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">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="c17">q1</font> )
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green" title="E94">A162</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:68_1_2_1_1_2_5_3_1"/><i><font color="Green" title="E95">U</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">q1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E93">A161</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green" title="E96">A164</font></i>: 
<font color="Maroon" title="c28">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="c17">q1</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="E5:68_1_2_1_1_2_5_3_1"/><i><font color="Green" title="E97">Y</font></i>: 
<font color="Maroon" title="c28">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="c25">MD1</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_3"><i><font color="Green" title="E91">A159</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E95">U</font></i></a></i>;<br/>
<a NAME="E6:68_1_2_1_1_2_5_3_1"/>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E25:68_1_2_1_1_2"><i><font color="Green" title="E74">A120</font></i></a>, <a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E7:68_1_2_1_1_2_5_3_1"/><b>then </b>
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:68_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green" title="E98">A165</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E98">A165</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E99">A166</font></i>: <font color="Maroon" title="c17">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E94">A162</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E96">A164</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a>, <a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a></i>
;<br/>
<a NAME="E11:68_1_2_1_1_2_5_3_1"/>
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="c25">MD1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_3_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_5_3_1_3"/>
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> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>hence </b><a NAME="E2:68_1_2_1_1_2_5_3_1_3"/>
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="c25">MD1</font>
 <b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_3"><i><font color="Green" title="E91">A159</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:68_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green" title="E100">A167</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="c25">MD1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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="c25">MD1</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 )
 
<b>by </b><i><a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<a NAME="E13:68_1_2_1_1_2_5_3_1"/>
<font color="Maroon" title="c25">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E14:68_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green" title="E101">A168</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c25">MD1</font>,1 <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>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E12:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E100">A167</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E15:68_1_2_1_1_2_5_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 
;<br/>
<a NAME="E16:68_1_2_1_1_2_5_3_1"/><b>then </b><i><font color="Green" title="E102">A169</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<a NAME="E17:68_1_2_1_1_2_5_3_1"/>
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>;<br/>
<a NAME="E18:68_1_2_1_1_2_5_3_1"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="integra1.html#E16:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E102">A169</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/>
<a NAME="E19:68_1_2_1_1_2_5_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E94">A162</font></i></a>, <a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E101">A168</font></i></a>, <a class="txt" href="integra1.html#E16:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E102">A169</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E97">Y</font></i></a></i>;<br/>
<b>hence </b><a NAME="E20:68_1_2_1_1_2_5_3_1"/>
<font color="Maroon" title="c17">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font>
 <b>by </b><i><a class="txt" href="integra1.html#E10:68_1_2_1_1_2_5_3_1"><i><font color="Green" title="E99">A166</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:68_1_2_1_1_2_5_3"/>
<font color="Maroon" title="c17">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font>
 <b>by </b><i><a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_3"><i><font color="Green" title="E92">A160</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:68_1_2_1_1_2_5"/><i><font color="Green" title="E91">A170</font></i>: 
<font color="Maroon" title="c19">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c26">MD2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4"><b>proof </b></a><div class="add">

<i><font color="Green" title="E92">A171</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5"><i><font color="Green" title="E81">A145</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span>

;<br/>
<i><font color="Green" title="E93">A172</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c26">MD2</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" 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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a></i>
<br/>.= 
<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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5"><i><font color="Green" title="E81">A145</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5"><i><font color="Green" title="E82">A146</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5"><i><font color="Green" title="E83">A147</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4"><i><font color="Green" title="E92">A171</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/>
<a NAME="E3:68_1_2_1_1_2_5_4"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">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="c19">q2</font> holds <br/><font color="Maroon" title="c19">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c26">MD2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c28">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:68_1_2_1_1_2_5_4_3"/><i><font color="Green" title="E94">A173</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">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="c19">q2</font> )
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green" title="E95">A174</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">q2</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="E3:68_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green" title="E96">A175</font></i>: 
<font color="Maroon" title="c28">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="c19">q2</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/>
<i><font color="Green" title="E97">A176</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
 <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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <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>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E15:68_1_2_1_1_2"><i><font color="Green" title="E68">A111</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>
;<br/>
<a NAME="E5:68_1_2_1_1_2_5_4_3"/><i><font color="Green" title="E98">A177</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E30:68_1_2_1_1_2"><i><font color="Green" title="E76">A122</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E96">A175</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></i>;<br/>
<a NAME="E6:68_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green" title="E99">A178</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:68_1_2_1_1_2_5_4_3"/><i><font color="Green" title="E100">A179</font></i>: 
 <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>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E8:68_1_2_1_1_2_5_4_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E97">A176</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E98">A177</font></i></a></i>;<br/>
<a NAME="E9:68_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green" title="E101">A180</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">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="E10:68_1_2_1_1_2_5_4_3"/><i><font color="Green" title="E102">A181</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> &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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E97">A176</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E98">A177</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E11:68_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green" title="E103">A182</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></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="E12:68_1_2_1_1_2_5_4_3"/><i><font color="Green" title="E104">A183</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <font color="Maroon" 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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ID</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="txt" href="integra1.html#E19:68_1_2_1_1_2"><i><font color="Green" title="E71">A117</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5"><i><font color="Green" title="E81">A145</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5"><i><font color="Green" title="E82">A146</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5"><i><font color="Green" title="E83">A147</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4"><i><font color="Green" title="E92">A171</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E13:68_1_2_1_1_2_5_4_3"/><b>then </b>
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E19:68_1_2_1_1_2"><i><font color="Green" title="E71">A117</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E94">A173</font></i></a>, <a class="txt" href="integra1.html#E11:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E103">A182</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:68_1_2_1_1_2_5_4_3"/><b>then </b><i><font color="Green" title="E105">A185</font></i>: 
<font color="Maroon" title="c28">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="c26">MD2</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/>
<i><font color="Green" title="E106">A186</font></i>: <font color="Maroon" title="c19">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E30:68_1_2_1_1_2"><i><font color="Green" title="E76">A122</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E96">A175</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c5">D2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E99">A178</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a>, <a class="txt" href="integra1.html#E9:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E101">A180</font></i></a></i>
;<br/>
<a NAME="E16:68_1_2_1_1_2_5_4_3"/><i><font color="Green" title="E107">A187</font></i>: 
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> &amp;  <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_2_1_1_2_5_4_3_3_1_1"/>
( <font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c28">n</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="68_1_2_1_1_2_5_4_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green" title="E108">A188</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green" title="E109">A189</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ;<br/><a NAME="E3:68_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green" title="E110">A190</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D2</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E97">A176</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E98">A177</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E100">A179</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E108">A188</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:68_1_2_1_1_2_5_4_3_3_1_1_1"/><i><font color="Green" title="E111">A191</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E24:68_1_2_1_1_2"><i><font color="Green" title="E73">A119</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><i><font color="Green" title="E112">A192</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="c26">MD2</font>,1</span>)</span> = 
 <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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E108">A188</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="integra1.html#E2:68_1_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E109">A189</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/><i><font color="Green" title="E113">A193</font></i>:  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,1</span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E108">A188</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="txt" href="integra1.html#E3:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E110">A190</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E7:68_1_2_1_1_2_5_4_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E114">A194</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E108">A188</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E112">A192</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/><i><font color="Green" title="E115">A195</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E108">A188</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5"><i><font color="Green" title="E84">A148</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E111">A191</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5"><i><font color="Green" title="E84">A148</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E111">A191</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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="c4">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="integra1.html#E1:68"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2"><i><font color="Green" title="E61">A104</font></i></a>, <a class="ref" href="integra1.html#D21" target="_self" title="INTEGRA1:def.21">Def21</a></i>
;<br/><b>hence </b><a NAME="E9:68_1_2_1_1_2_5_4_3_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E108">A188</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E112">A192</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E113">A193</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/><b>thus </b><a NAME="E10:68_1_2_1_1_2_5_4_3_3_1_1_1"/>
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> &amp;  <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E114">A194</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5_4_3_3_1_1_1"><i><font color="Green" title="E115">A195</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self" title="INTEGRA1:th.10">Th10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E108">A196</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E109">A197</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E16:68_1_2_1_1_2"><i><font color="Green" title="E69">A112</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>consider </b><font color="Maroon" title="c29">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E110">A198</font></i>: 
<font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c29">n1</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E94">A173</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c29">n1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E111">A199</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="integra1.html#E10:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E102">A181</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E110">A198</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E7:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E112">A200</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c28">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">MD2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
not <font color="Maroon" title="c28">n</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E94">A173</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E108">A196</font></i></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<a NAME="E2:68_1_2_1_1_2_5_4_3_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#T31" title="NAT_2:th.31">NAT_2:31</a></i>;<br/>
<b>hence </b><a NAME="E3:68_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon" title="c28">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>

<a NAME="E4:68_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</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#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E5:68_1_2_1_1_2_5_4_3_3_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c28">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E6:68_1_2_1_1_2_5_4_3_3_1_1_2_1"/>
<font color="Maroon" title="c28">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">MD2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E19:68_1_2_1_1_2"><i><font color="Green" title="E71">A117</font></i></a>, <a class="txt" href="integra1.html#E11:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E103">A182</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E104">A183</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E113">A201</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3_1_1_2_2"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>
<br/>.= 
<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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font>

;<br/>
<b>hence </b><a NAME="E2:68_1_2_1_1_2_5_4_3_3_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E9:68_1_2_1_1_2_5_4_3_3_1_1_2"/><i><font color="Green" title="E114">A202</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_4_3_3_1_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:68_1_2_1_1_2_5_4_3_3_1_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:68_1_2_1_1_2_5_4_3_3_1_1_2_3"/><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="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c28">n</font>
 
;<br/>
<a NAME="E3:68_1_2_1_1_2_5_4_3_3_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c28">n</font> <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> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E24:68_1_2_1_1_2"><i><font color="Green" title="E73">A119</font></i></a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E4:68_1_2_1_1_2_5_4_3_3_1_1_2_3"/><b>then </b>
<font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:68_1_2_1_1_2_5_4_3_3_1_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E94">A173</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>


</div><b>end;</b></div><i><font color="Green" title="E115">A203</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="c26">MD2</font>,<font color="Maroon" title="c28">n</font></span>)</span> = 
<font color="Maroon" title="c26">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E108">A196</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17" title="INTEGRA1:func.17">indx</a> <font color="Maroon" title="c5">D2</font>,<font color="Maroon" title="c4">D1</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5"><i><font color="Green" title="E82">A146</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5"><i><font color="Green" title="E83">A147</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E109">A197</font></i></a>, <a class="txt" href="integra1.html#E4:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E110">A198</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E111">A199</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E112">A200</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><i><font color="Green" title="E116">A204</font></i>:  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</font></span>)</span> = 
<font color="Maroon" title="c26">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E108">A196</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span>
<b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2"><i><font color="Green" title="E67">A110</font></i></a>, <a class="txt" href="integra1.html#E19:68_1_2_1_1_2"><i><font color="Green" title="E71">A117</font></i></a>, <a class="txt" href="integra1.html#E5:68_1_2_1_1_2_5"><i><font color="Green" title="E82">A146</font></i></a>, <a class="txt" href="integra1.html#E6:68_1_2_1_1_2_5"><i><font color="Green" title="E83">A147</font></i></a>, <a class="txt" href="integra1.html#E1:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E94">A173</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E95">A174</font></i></a>, <a class="txt" href="integra1.html#E11:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E103">A182</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E104">A183</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E109">A197</font></i></a>, <a class="txt" href="integra1.html#E8:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E113">A201</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
;<br/><a NAME="E12:68_1_2_1_1_2_5_4_3_3_1_1_2"/><b>then </b><i><font color="Green" title="E117">A205</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</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="txt" href="integra1.html#E10:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E115">A203</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/><i><font color="Green" title="E118">A206</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</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" target="_self" title="INTEGRA1:th.5">Th5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c5">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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</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="txt" href="integra1.html#E9:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E101">A180</font></i></a>, <a class="txt" href="integra1.html#E9:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E114">A202</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
<br/>.= 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">D2</font> <a href="seq_1.html#K1" title="SEQ_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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</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="txt" href="integra1.html#E9:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E101">A180</font></i></a>, <a class="txt" href="integra1.html#E9:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E114">A202</font></i></a>, <a class="ref" href="integra1.html#D5" target="_self" title="INTEGRA1:def.5">Def5</a></i>
;<br/><b>hence </b><a NAME="E14:68_1_2_1_1_2_5_4_3_3_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E10:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E115">A203</font></i></a>, <a class="txt" href="integra1.html#E11:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E116">A204</font></i></a>, <a class="ref" href="integra1.html#T5" target="_self" title="INTEGRA1:th.5">Th5</a></i>;<br/><b>thus </b><a NAME="E15:68_1_2_1_1_2_5_4_3_3_1_1_2"/>
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> &amp;  <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a>, <a class="txt" href="integra1.html#E12:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E117">A205</font></i></a>, <a class="txt" href="integra1.html#E13:68_1_2_1_1_2_5_4_3_3_1_1_2"><i><font color="Green" title="E118">A206</font></i></a>, <a class="ref" href="integra1.html#T10" target="_self" title="INTEGRA1:th.10">Th10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:68_1_2_1_1_2_5_4_3_3"/>
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</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"><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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> &amp;  <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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E17:68_1_2_1_1_2_5_4_3"/><b>then </b>
<font color="Maroon" title="c27">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c26">MD2</font>,<font color="Maroon" title="c28">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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="c6">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c28">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a></i>;<br/>
<b>hence </b><a NAME="E18:68_1_2_1_1_2_5_4_3"/>
<font color="Maroon" title="c19">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c26">MD2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">n</font>
 <b>by </b><i><a class="txt" href="integra1.html#E15:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E106">A186</font></i></a>, <a class="txt" href="integra1.html#E16:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E107">A187</font></i></a>, <a class="ref" href="integra1.html#D8" target="_self" title="INTEGRA1:def.8">Def8</a>, <a class="txt" href="integra1.html#E14:68_1_2_1_1_2_5_4_3"><i><font color="Green" title="E105">A185</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68_1_2_1_1_2_5_4"/>
<font color="Maroon" title="c19">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c26">MD2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E30:68_1_2_1_1_2"><i><font color="Green" title="E76">A122</font></i></a>, <a class="txt" href="integra1.html#E2:68_1_2_1_1_2_5_4"><i><font color="Green" title="E93">A172</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:68_1_2_1_1_2_5"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c25">MD1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c26">MD2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1_1_2_5_5"><b>proof </b></a><div class="add">

<a NAME="E1:68_1_2_1_1_2_5_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra1.html#E4:68_1_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:68_1_2_1_1_2_5_5"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:68_1_2_1_1_2_5_5"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">MD1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c25">MD1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c26">MD2</font> )
 <b>by </b><i><a class="ref" href="integra1.html#T31" target="_self" title="INTEGRA1:th.31">Th31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:68_1_2_1_1_2_5"/><b>then </b>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c25">MD1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c27">g</font>,<font color="Maroon" title="c26">MD2</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E17:68_1_2_1_1_2_5"><i><font color="Green" title="E88">A152</font></i></a>, <a class="txt" href="integra1.html#E18:68_1_2_1_1_2_5"><i><font color="Green" title="E89">A157</font></i></a>, <a class="ref" href="integra1.html#T33" target="_self" title="INTEGRA1:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E23:68_1_2_1_1_2_5"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">q2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E19:68_1_2_1_1_2_5"><i><font color="Green" title="E90">A158</font></i></a>, <a class="txt" href="integra1.html#E20:68_1_2_1_1_2_5"><i><font color="Green" title="E91">A170</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E34:68_1_2_1_1_2"/><i><font color="Green" title="E80">A207</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c16">p1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">q1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E27:68_1_2_1_1_2"><i><font color="Green" title="E75">A121</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/><a NAME="E35:68_1_2_1_1_2"/><i><font color="Green" title="E81">A208</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c18">p2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">q2</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E30:68_1_2_1_1_2"><i><font color="Green" title="E76">A122</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/><a NAME="E36:68_1_2_1_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c16">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E34:68_1_2_1_1_2"><i><font color="Green" title="E80">A207</font></i></a></i>;<br/><a NAME="E37:68_1_2_1_1_2"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c16">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E33:68_1_2_1_1_2"><i><font color="Green" title="E79">A143</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E38:68_1_2_1_1_2"/><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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c16">p1</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="E39:68_1_2_1_1_2"/><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"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c19">q2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c18">p2</font>
 <b>by </b><i><a class="txt" href="integra1.html#E1:68_1_2"><i><font color="Green" title="E53">A95</font></i></a>, <a class="txt" href="integra1.html#E7:68_1_2_1_1_2"><i><font color="Green" title="E60">A103</font></i></a>, <a class="txt" href="integra1.html#E31:68_1_2_1_1_2"><i><font color="Green" title="E77">A123</font></i></a>, <a class="txt" href="integra1.html#E32:68_1_2_1_1_2"><i><font color="Green" title="E78">A133</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</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="E40:68_1_2_1_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E35:68_1_2_1_1_2"><i><font color="Green" title="E81">A208</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:68_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:68_1"/>
for <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="binarith.html#S1" title="BINARITH:sch.1">BINARITH:sch 1</a>(<a class="txt" href="integra1.html#E1:68_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="integra1.html#E2:68_1"><i><font color="Green" title="E52">A94</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:68"/>
for <font color="Olive" title="b1">i</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D1</font> holds <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="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">D1</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="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="Olive" title="b1">i</font></span>)</span></span>)</span>
 ;<br/>


</div>
