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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">f</font> being   <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>  st <font color="Olive" title="b1">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Olive" title="b2">D2</font> &amp; <font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>  holds <br/> for <font color="Olive" title="b4">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="b4">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="Olive" title="b1">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b4">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Olive" title="b2">D2</font>,<font color="Olive" title="b1">D1</font>,<font color="Olive" title="b4">i</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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>  st <font color="Maroon" title="c2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>  holds <br/> for <font color="Olive" title="b2">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="b2">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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Olive" title="b2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Olive" title="b2">i</font>)</span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a>  implies  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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c2">D1</font> <a href="integra1.html#R1" title="INTEGRA1:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font>
 <span class="kw">and </span><br/><a NAME="E2:61"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">A</font> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span></span><br/>

<a NAME="E3:61"/>
 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="nat_1.html#NM1" title="NAT_1:NM.1">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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>( $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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,$1)</span>)</span></span>)</span> );<br/>
<a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">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="c2">D1</font>,1)</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B</font> =  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">DD2</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">DD1</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c4">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="c2">D1</font>,1)</span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T10" title="PARTFUN1:th.10">PARTFUN1:10</a></span>;<br/>
<a NAME="E2:61_1_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:61_1_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
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="c2">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><br/>

<a NAME="E4:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E5:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<a NAME="E6:61_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T11" title="SEQ_4:th.11">SEQ_4:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">DD1</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1) as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E4:61_1_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="integra1.html#T37" target="_self" title="INTEGRA1:th.37">Th39</a></span>;<br/>
<a NAME="E8:61_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A8</font></span>: 
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="c2">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E10:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">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="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<span class="lab"><font color="Green" title="E53">A10</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/>
<a NAME="E12:61_1_1"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E13:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">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="c2">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E14:61_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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">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="c2">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="61_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> implies <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c11">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_1_2"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_1_2"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c11">k</font></span><br/>

<a NAME="E3:61_1_1_2"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_1_1"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E1:61_1_1_2"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E2:61_1_1_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:61_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:61_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<a NAME="E6:61_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E7:61_1_1_2"/><span class="lab"><font color="Green" title="E59">A16</font></span>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_1"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E1:61_1_1_2"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E2:61_1_1_2"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E8:61_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:61_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c11">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_2"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="txt" href="#E6:61_1_1_2"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A17</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_1_1"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E16:61_1_1"/><span class="lab"><font color="Green" title="E57">A18</font></span>: 
 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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c10">DD1</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="61_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c10">DD1</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c11">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c11">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_1_3"/><span class="lab"><font color="Green" title="E58">A19</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_1_3"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c11">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c10">DD1</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c11">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c11">i</font></span><br/>

<a NAME="E3:61_1_1_3"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:61_1_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E5:61_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A22</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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="c2">D1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E62">A23</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>
;<br/>
<a NAME="E8:61_1_1_3"/>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E9:61_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A24</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A25</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c11">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c11">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E1:61_1_1_3"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="txt" href="#E2:61_1_1_3"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_3"><span class="lab"><font color="Green" title="E62">A23</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c4">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="c2">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="c2">D1</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E65">A26</font></span>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E66">A27</font></span>: <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c11">i</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c10">DD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="txt" href="#E1:61_1_1_3"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="txt" href="#E2:61_1_1_3"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c9">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1_3"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
;<br/>
 <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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1_3"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1_3"><span class="lab"><font color="Green" title="E63">A24</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E15:61_1_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_3"><span class="lab"><font color="Green" title="E61">A22</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:61_1_1_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c11">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E12:61_1_1_3"><span class="lab"><font color="Green" title="E66">A27</font></span></a>, <a class="txt" href="#E11:61_1_1_3"><span class="lab"><font color="Green" title="E65">A26</font></span></a>, <a class="txt" href="#E10:61_1_1_3"><span class="lab"><font color="Green" title="E64">A25</font></span></a>, <a class="ref" href="relat_1.html#T68" title="RELAT_1:th.68">RELAT_1:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E17:61_1_1"/><span class="lab"><font color="Green" title="E58">A28</font></span>: 
<font color="Maroon" title="c9">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="c2">D1</font>,1)</span>)</span> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c11">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> <span class="kw"> such that </span><br/><a NAME="E2:61_1_1_4"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="rfunct_1.html#T71" title="RFUNCT_1:th.71">RFUNCT_1:71</a></span>;<br/>
<a NAME="E3:61_1_1_4"/>
 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="c2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font></span>)</span> holds <br/><font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">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="61_1_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">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="c2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font></span>)</span> implies <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )</span><br/>

<a NAME="E1:61_1_1_4_1"/><span class="lab"><font color="Green" title="E60">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:61_1_1_4_1"/>
<font color="Maroon" title="c12">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="c2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<a NAME="E3:61_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A31</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:61_1_1_4_1"/><span class="lab"><font color="Green" title="E62">A32</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_4_1"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E1:61_1_1_4_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:61_1_1_4_1"/>
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_4"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="txt" href="#E3:61_1_1_4_1"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="txt" href="#E4:61_1_1_4_1"><span class="lab"><font color="Green" title="E62">A32</font></span></a>, <a class="txt" href="#E1:61_1_1_4_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:61_1_1_4_1"/>
<font color="Maroon" title="c11">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_4_1"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:61_1_1_4"/>
<font color="Maroon" title="c9">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="c2">D1</font>,1)</span>)</span> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T71" title="RFUNCT_1:th.71">RFUNCT_1:71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:61_1_1"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></span>;<br/>
<a NAME="E19:61_1_1"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<a NAME="E20:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A35</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E21:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A36</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E22:61_1_1"/><span class="lab"><font color="Green" title="E63">A37</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61_1_1"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E23:61_1_1"/><span class="kw">then </span>
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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:61_1_1"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:61_1_1"/><span class="kw">then </span>
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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E25:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A38</font></span>: 
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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E26:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E27:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61_1_1"><span class="lab"><font color="Green" title="E59">A33</font></span></a></span>;<br/>
<a NAME="E28:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E29:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<a NAME="E30:61_1_1"/><span class="kw">then </span>
 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="c3">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <font color="Maroon" title="c1">A</font> )
 
;<br/>
<a NAME="E31:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">DD2</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>) as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61_1_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E21:61_1_1"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="txt" href="#E25:61_1_1"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E5:61_1_1"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="integra1.html#T37" target="_self" title="INTEGRA1:th.37">Th39</a></span>;<br/>
<a NAME="E33:61_1_1"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<a NAME="E34:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A39</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E35:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E36:61_1_1"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:61_1_1"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E37:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A42</font></span>: 
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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:61_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E38:61_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:61_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E39:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A43</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:61_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E40:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A44</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E41:61_1_1"/><span class="lab"><font color="Green" title="E71">A45</font></span>: 
 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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="61_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c12">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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> implies <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_1_5"/><span class="lab"><font color="Green" title="E72">A46</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_1_5"/><span class="lab"><font color="Green" title="E73">A47</font></span>: 
<font color="Maroon" title="c12">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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/>

<a NAME="E3:61_1_1_5"/><span class="lab"><font color="Green" title="E74">A48</font></span>: 
<font color="Maroon" title="c12">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="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:61_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="txt" href="#E1:61_1_1_5"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="txt" href="#E2:61_1_1_5"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:61_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:61_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<a NAME="E6:61_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A49</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/>
<a NAME="E7:61_1_1_5"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c12">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:61_1_1"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="txt" href="#E37:61_1_1"><span class="lab"><font color="Green" title="E68">A42</font></span></a>, <a class="txt" href="#E1:61_1_1_5"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="txt" href="#E2:61_1_1_5"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E3:61_1_1_5"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E8:61_1_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:61_1_1_5"/>
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_1_1_5"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E42:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A50</font></span>: 
 <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:61_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E43:61_1_1"/><span class="lab"><font color="Green" title="E73">A51</font></span>: 
 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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c11">DD2</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="61_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c11">DD2</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_1_6"/><span class="lab"><font color="Green" title="E74">A52</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_1_6"/><span class="lab"><font color="Green" title="E75">A53</font></span>: 
<font color="Maroon" title="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c11">DD2</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E3:61_1_1_6"/><span class="lab"><font color="Green" title="E76">A54</font></span>: 
<font color="Maroon" title="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_6"><span class="lab"><font color="Green" title="E75">A53</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E4:61_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A55</font></span>: 
<font color="Maroon" title="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:61_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A56</font></span>: 
<font color="Maroon" title="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:61_1_1_6"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:61_1_1_6_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E2:61_1_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:61_1_1_6"><span class="lab"><font color="Green" title="E77">A55</font></span></a></span>;<br/>
<a NAME="E3:61_1_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A57</font></span>: 
<font color="Maroon" title="c12">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1_6_1_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">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="61_1_1_6_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_6_1_1_1_1"/><span class="lab"><font color="Green" title="E80">A58</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span><br/><div class="add"><a NAME="E2:61_1_1_6_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A59</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:61_1_1"><span class="lab"><font color="Green" title="E72">A50</font></span></a>, <a class="txt" href="#E5:61_1_1_6"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E3:61_1_1_6_1_1_1_1"/><span class="kw">then </span>
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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E4:61_1_1_6_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A60</font></span>: 
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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="lab"><font color="Green" title="E83">A61</font></span>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E82">A60</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E82">A60</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/> <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:61_1_1"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="txt" href="#E3:61_1_1_6"><span class="lab"><font color="Green" title="E76">A54</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E80">A58</font></span></a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E81">A59</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:61_1_1_6_1_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E83">A61</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E80">A62</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span><br/><div class="add"><a NAME="E2:61_1_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E81">A63</font></span>: 
<font color="Maroon" title="c12">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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:61_1_1_6_1_1_1_2_1"/>
 not <font color="Maroon" title="c12">i</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E80">A62</font></span></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></span>;<br/>
<a NAME="E2:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A64</font></span>: 
<font color="Maroon" title="c12">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></span>;<br/>
<a NAME="E3:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A65</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E4:61_1_1_6_1_1_1_2_1"/><span class="lab"><font color="Green" title="E84">A66</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c12">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="Olive" title="b1">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<a NAME="E5:61_1_1_6_1_1_1_2_1"/><span class="lab"><font color="Green" title="E85">A67</font></span>: 
<font color="Maroon" title="c12">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E3:61_1_1_6"><span class="lab"><font color="Green" title="E76">A54</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/>
<a NAME="E6:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61_1_1"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_6_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A66</font></span></a>, <a class="txt" href="#E3:61_1_1_6_1_1_1_2_1"><span class="lab"><font color="Green" title="E83">A65</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A68</font></span>: 
<font color="Maroon" title="c12">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:61_1_1_6_1_1_1_2_1"/>
<font color="Maroon" title="c12">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_6_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A64</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E10:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_6_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A66</font></span></a>, <a class="txt" href="#E5:61_1_1_6_1_1_1_2_1"><span class="lab"><font color="Green" title="E85">A67</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E11:61_1_1_6_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_1_6_1_1_1_2_1"><span class="lab"><font color="Green" title="E86">A68</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:61_1_1_6_1_1_1_2_1"/>
<font color="Maroon" title="c12">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#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></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="c12">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="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:61_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="txt" href="#E41:61_1_1"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
;<br/><a NAME="E4:61_1_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A69</font></span>: 
<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="c12">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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E81">A63</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><a NAME="E5:61_1_1_6_1_1_1_2"/>
<font color="Maroon" title="c12">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E22:61_1_1"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="txt" href="#E3:61_1_1_6"><span class="lab"><font color="Green" title="E76">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:61_1_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E7:61_1_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:61_1_1_6_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:61_1_1_6"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E9:61_1_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A70</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:61_1_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="txt" href="#E41:61_1_1"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
;<br/><a NAME="E11:61_1_1_6_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A71</font></span>: 
<font color="Maroon" title="c11">DD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E83">A70</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>;<br/><span class="lab"><font color="Green" title="E85">A72</font></span>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_6_1"><span class="lab"><font color="Green" title="E79">A57</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E80">A62</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_6_1"><span class="lab"><font color="Green" title="E79">A57</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E80">A62</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/> <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c12">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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="c12">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="c12">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</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="c12">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E80">A62</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c12">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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="txt" href="#E1:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E80">A62</font></span></a>, <a class="txt" href="#E4:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="txt" href="#E11:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E84">A71</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:61_1_1_6_1_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E85">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:61_1_1_6_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c11">DD2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:61_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A73</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c9">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="c3">D2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6"><span class="lab"><font color="Green" title="E78">A56</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E8:61_1_1_6"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E9:61_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:61_1_1_6"><span class="lab"><font color="Green" title="E77">A55</font></span></a></span>;<br/>
<a NAME="E10:61_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A74</font></span>: 
<font color="Maroon" title="c12">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:61_1_1_6"/><span class="lab"><font color="Green" title="E81">A75</font></span>: 
<font color="Maroon" title="c12">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_1_6"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E82">A76</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1_6_2_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">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="61_1_1_6_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_6_2_1_1"/><span class="lab"><font color="Green" title="E83">A77</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/><div class="add"><a NAME="E2:61_1_1_6_2_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:61_1_1"><span class="lab"><font color="Green" title="E72">A50</font></span></a>, <a class="txt" href="#E11:61_1_1_6"><span class="lab"><font color="Green" title="E81">A75</font></span></a>, <a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/><a NAME="E3:61_1_1_6_2_1_1"/><span class="kw">then </span>
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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/><a NAME="E4:61_1_1_6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A78</font></span>: 
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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:61_1_1_6_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61_1_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E21:61_1_1"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E6:61_1_1_6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A79</font></span>: 
<font color="Maroon" title="c3">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="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/><a NAME="E7:61_1_1_6_2_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E83">A77</font></span></a>, <a class="txt" href="#E4:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E84">A78</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E8:61_1_1_6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A80</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E9:61_1_1_6_2_1_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E83">A77</font></span></a>, <a class="txt" href="#E4:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E84">A78</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E10:61_1_1_6_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A81</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E6:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E85">A79</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E11:61_1_1_6_2_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T11" title="SEQ_4:th.11">SEQ_4:11</a></span>;<br/><span class="kw">hence </span><a NAME="E12:61_1_1_6_2_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E86">A80</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/><a NAME="E13:61_1_1_6_2_1_1"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T11" title="SEQ_4:th.11">SEQ_4:11</a></span>;<br/><a NAME="E14:61_1_1_6_2_1_1"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E86">A80</font></span></a>, <a class="txt" href="#E10:61_1_1_6_2_1_1"><span class="lab"><font color="Green" title="E87">A81</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:61_1_1_6_2_1_1"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E83">A82</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp;  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )</span><br/><div class="add"><a NAME="E2:61_1_1_6_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">i</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></span>;<br/><a NAME="E3:61_1_1_6_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></span>;<br/><a NAME="E4:61_1_1_6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A83</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E5:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E85">A84</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c12">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="Olive" title="b1">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_6"><span class="lab"><font color="Green" title="E74">A52</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E6:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E86">A85</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D2" target="_self" title="INTEGRA1:def.2">Def2</a></span>;<br/><a NAME="E7:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E87">A86</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1_6"><span class="lab"><font color="Green" title="E80">A74</font></span></a>, <a class="txt" href="#E1:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E8:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E88">A87</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E9:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E89">A88</font></span>: 
<font color="Maroon" title="c12">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E3:61_1_1_6"><span class="lab"><font color="Green" title="E76">A54</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/><a NAME="E10:61_1_1_6_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61_1_1"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:61_1_1_6_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E85">A84</font></span></a>, <a class="txt" href="#E4:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E12:61_1_1_6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A89</font></span>: 
<font color="Maroon" title="c12">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E13:61_1_1_6_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E14:61_1_1_6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A90</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E87">A86</font></span></a>, <a class="txt" href="#E8:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E6:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/><a NAME="E15:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E92">A91</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E16:61_1_1_6_2_1_2"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:61_1_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E9:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E89">A88</font></span></a>, <a class="txt" href="#E12:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E17:61_1_1_6_2_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E7:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E87">A86</font></span></a>, <a class="txt" href="#E15:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E92">A91</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/><a NAME="E18:61_1_1_6_2_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E91">A90</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E19:61_1_1_6_2_1_2"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span><br/><a NAME="E20:61_1_1_6_2_1_2"/><span class="lab"><font color="Green" title="E93">A92</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1_6"><span class="lab"><font color="Green" title="E80">A74</font></span></a>, <a class="txt" href="#E1:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/><a NAME="E21:61_1_1_6_2_1_2"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_1_6"><span class="lab"><font color="Green" title="E80">A74</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E22:61_1_1_6_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A93</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E93">A92</font></span></a>, <a class="txt" href="#E8:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E6:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></span>;<br/><a NAME="E23:61_1_1_6_2_1_2"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E19:61_1_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E3:61_1_1_6"><span class="lab"><font color="Green" title="E76">A54</font></span></a>, <a class="txt" href="#E10:61_1_1_6"><span class="lab"><font color="Green" title="E80">A74</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/><a NAME="E24:61_1_1_6_2_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E20:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E93">A92</font></span></a>, <a class="txt" href="#E15:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E92">A91</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/><a NAME="E25:61_1_1_6_2_1_2"/><span class="kw">then </span>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61_1_1_6_2_1_2"><span class="lab"><font color="Green" title="E94">A93</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E26:61_1_1_6_2_1_2"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E13:61_1_1_6"/><span class="lab"><font color="Green" title="E83">A94</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E14:61_1_1_6"/><span class="lab"><font color="Green" title="E84">A95</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E15:61_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:61_1_1_6"><span class="lab"><font color="Green" title="E77">A55</font></span></a></span>;<br/>
<a NAME="E16:61_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A96</font></span>: 
<font color="Maroon" title="c12">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E17:61_1_1_6"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E18:61_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A97</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c12">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="c2">D1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:61_1_1_6"><span class="lab"><font color="Green" title="E83">A94</font></span></a>, <a class="txt" href="#E12:61_1_1_6"><span class="lab"><font color="Green" title="E82">A76</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/>
<span class="lab"><font color="Green" title="E87">A98</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>
<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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_1_6"><span class="lab"><font color="Green" title="E84">A95</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:61_1_1_6"><span class="lab"><font color="Green" title="E77">A55</font></span></a>, <a class="txt" href="#E19:61_1_1_6"><span class="lab"><font color="Green" title="E87">A98</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c4">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="c3">D2</font>,<font color="Maroon" title="c12">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="c3">D2</font>,<font color="Maroon" title="c12">i</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_1_6"><span class="lab"><font color="Green" title="E85">A96</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E21:61_1_1_6"/>
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_6"><span class="lab"><font color="Green" title="E79">A73</font></span></a>, <a class="txt" href="#E18:61_1_1_6"><span class="lab"><font color="Green" title="E86">A97</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E44:61_1_1"/>
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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_1"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E45:61_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E46:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A99</font></span>: 
 <a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E47:61_1_1"/><span class="lab"><font color="Green" title="E75">A100</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:61_1_1"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E48:61_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E49:61_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A101</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c9">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:61_1_1"><span class="lab"><font color="Green" title="E75">A100</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E50:61_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">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="c2">D1</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E51:61_1_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">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="c2">D1</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="integra1.html#T8" target="_self" title="INTEGRA1:th.8">Th10</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E52:61_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">g</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E53:61_1_1"/><span class="kw">then </span>
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c9">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="c9">g</font>,<font color="Maroon" title="c11">DD2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_1"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="txt" href="#E17:61_1_1"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="integra1.html#T31" target="_self" title="INTEGRA1:th.31">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E54:61_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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,1)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:61_1_1"><span class="lab"><font color="Green" title="E74">A99</font></span></a>, <a class="txt" href="#E49:61_1_1"><span class="lab"><font color="Green" title="E76">A101</font></span></a>, <a class="txt" href="#E43:61_1_1"><span class="lab"><font color="Green" title="E73">A51</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:61_1"/><span class="lab"><font color="Green" title="E47">A102</font></span>: 
 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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:61_1_2"/><span class="lab"><font color="Green" title="E48">A103</font></span>: 
( <font color="Maroon" title="c5">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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">assume </span><a NAME="E2:61_1_2"/><span class="lab"><font color="Green" title="E49">A104</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span><br/>

<a NAME="E3:61_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A105</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E4:61_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A106</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E5:61_1_2"/><span class="lab"><font color="Green" title="E52">A107</font></span>: 
<font color="Maroon" title="c5">k</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2"><span class="lab"><font color="Green" title="E50">A105</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_2_1_1_1"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:61_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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E53">A108</font></span>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c6">IDK</font> =  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>);<br/><span class="kw">set </span><font color="Maroon" title="c7">IDK1</font> =  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c8">K1D2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c9">KD1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">K1D1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/><a NAME="E2:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E54">A109</font></span>: 
<font color="Maroon" title="c5">k</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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2"><span class="lab"><font color="Green" title="E52">A107</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/><a NAME="E3:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A110</font></span>: 
 <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p1</font>, <font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E5:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A111</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">and </span><br/><a NAME="E6:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A112</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E7:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E58">A113</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c13">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T23" title="FINSEQ_2:th.23">FINSEQ_2:23</a></span>;<br/><a NAME="E8:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A114</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E9:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A115</font></span>: 
<font color="Maroon" title="c5">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="c2">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2"><span class="lab"><font color="Green" title="E52">A107</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E10:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A116</font></span>: 
<font color="Maroon" title="c5">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/><a NAME="E11:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A117</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A111</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><a NAME="E12:61_1_2_1_1_2"/>
 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="c12">p1</font> holds <br/><font color="Maroon" title="c12">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">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="61_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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="c12">p1</font> implies <font color="Maroon" title="c12">p1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A118</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A119</font></span>: 
<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="c12">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">p1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span><br/>

<a NAME="E3:61_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E65">A120</font></span>: 
<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="c12">p1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E63">A118</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E64">A119</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:61_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A121</font></span>: 
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A117</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:61_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A122</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<a NAME="E6:61_1_2_1_1_2_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E7:61_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A123</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<span class="lab"><font color="Green" title="E69">A124</font></span>:  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A109</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>
;<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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A116</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>
;<br/>
<a NAME="E10:61_1_2_1_1_2_1"/><span class="kw">then </span>
<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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E66">A121</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E68">A123</font></span></a>, <a class="txt" href="#E8:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E69">A124</font></span></a></span>;<br/>
<a NAME="E11:61_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A125</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A126</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E70">A125</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<span class="lab"><font color="Green" title="E72">A127</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</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"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font></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="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E67">A122</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<a NAME="E14:61_1_2_1_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="c12">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E65">A120</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:61_1_2_1_1_2_1"/>
<font color="Maroon" title="c12">p1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E58">A113</font></span></a>, <a class="txt" href="#E13:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E72">A127</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E71">A126</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A128</font></span>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">D1</font>)</span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A117</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><a NAME="E14:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E64">A129</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/><a NAME="E15:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A130</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E16:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A131</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E17:61_1_2_1_1_2"/>
 not <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></span>;<br/><a NAME="E18:61_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></span>;<br/><a NAME="E19:61_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E20:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A132</font></span>: 
<font color="Maroon" title="c5">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="c2">D1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A115</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E21:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A133</font></span>: 
<font color="Maroon" title="c5">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="c2">D1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E22:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A134</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/><a NAME="E23:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E70">A135</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:61_1_2_1_1_2_2"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E2:61_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A136</font></span>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:61_1_2_1_1_2_2"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:61_1_2_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A137</font></span>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A134</font></span></a>, <a class="txt" href="#E14:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A129</font></span></a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E5:61_1_2_1_1_2_2"/>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:61_1_2_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E72">A137</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_2"><span class="lab"><font color="Green" title="E71">A136</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E24:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A138</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_2_1_1_2_3"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:61_1_2_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A139</font></span>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A134</font></span></a>, <a class="txt" href="#E14:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A129</font></span></a>, <a class="ref" href="seqm_3.html#D1" title="SEQM_3:def.1">SEQM_3:def 1</a></span>;<br/>
<a NAME="E3:61_1_2_1_1_2_3"/>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<a NAME="E4:61_1_2_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_3"><span class="lab"><font color="Green" title="E72">A139</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:61_1_2_1_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">ID</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E26:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E72">A140</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c14">ID</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">ID</font> = <font color="Maroon" title="c14">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E28:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E73">A141</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/><a NAME="E29:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A142</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A130</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E30:61_1_2_1_1_2"/><span class="kw">then </span>
 <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">p2</font>, <font color="Maroon" title="c17">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> <span class="kw"> such that </span><br/><a NAME="E32:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E75">A143</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)
 <span class="kw">and </span><br/><a NAME="E33:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E76">A144</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E34:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A145</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p2</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c17">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="ref" href="finseq_2.html#T23" title="FINSEQ_2:th.23">FINSEQ_2:23</a></span>;<br/><a NAME="E35:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E78">A146</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A130</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E36:61_1_2_1_1_2"/>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/><a NAME="E37:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A147</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E38:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A148</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E39:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E81">A149</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">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="c17">q2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c18">MD2</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c19">MD1</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c20">DD1</font> =  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c4">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>;<br/>
<a NAME="E1:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E82">A150</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">g</font> = <font color="Maroon" title="c4">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T10" title="PARTFUN1:th.10">PARTFUN1:10</a></span>;<br/>
<a NAME="E3:61_1_2_1_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 
;<br/>
<a NAME="E4:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A151</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E5:61_1_2_1_1_2_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" title="RELAT_1:th.61">RELAT_1:61</a></span>;<br/>
<a NAME="E6:61_1_2_1_1_2_4"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:61_1_2_1_1_2_4"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="ref" href="integra1.html#T8" target="_self" title="INTEGRA1:th.8">Th10</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E8:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A152</font></span>: 
<font color="Maroon" title="c22">g</font> is  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E9:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E85">A153</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E10:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E86">A154</font></span>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<a NAME="E11:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E87">A155</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A135</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A156</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:61_1_2_1_1_2_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E82">A150</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E14:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A157</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:61_1_2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E69">A134</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E16:61_1_2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E83">A151</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">MD2</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>) as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E64">A129</font></span></a>, <a class="txt" href="#E11:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E87">A155</font></span></a>, <a class="txt" href="#E14:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E89">A157</font></span></a>, <a class="txt" href="#E10:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E86">A154</font></span></a>, <a class="txt" href="#E9:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E85">A153</font></span></a>, <a class="ref" href="integra1.html#T37" target="_self" title="INTEGRA1:th.37">Th39</a></span>;<br/>
<span class="lab"><font color="Green" title="E90">A158</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E87">A155</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>

;<br/>
<a NAME="E19:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E91">A159</font></span>: 
 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">q2</font> holds <br/><font color="Maroon" title="c17">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">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="61_1_2_1_1_2_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <font color="Maroon" title="c24">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">q2</font> implies <font color="Maroon" title="c17">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E92">A160</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E93">A161</font></span>: 
<font color="Maroon" title="c24">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">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font></span><br/>

<a NAME="E3:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E94">A162</font></span>: 
<font color="Maroon" title="c24">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">q2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E92">A160</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E93">A161</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:61_1_2_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A163</font></span>: 
<font color="Maroon" title="c24">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">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:61_1_2_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A164</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A143</font></span></a>, <a class="txt" href="#E34:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A145</font></span></a>, <a class="ref" href="finseq_1.html#T28" title="FINSEQ_1:th.28">FINSEQ_1:28</a></span>;<br/>
<a NAME="E6:61_1_2_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A165</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<a NAME="E7:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E98">A166</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ID</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A131</font></span></a>, <a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="txt" href="#E26:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="txt" href="#E11:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E87">A155</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E88">A156</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E82">A150</font></span></a>, <a class="txt" href="#E18:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E90">A158</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<span class="lab"><font color="Green" title="E99">A167</font></span>:  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A142</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>
;<br/>
<a NAME="E9:61_1_2_1_1_2_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E96">A164</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E10:61_1_2_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A168</font></span>: 
<font color="Maroon" title="c24">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E11:61_1_2_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E92">A160</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E98">A166</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E12:61_1_2_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A169</font></span>: 
<font color="Maroon" title="c24">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="c23">MD2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E13:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E102">A170</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/>
<a NAME="E14:61_1_2_1_1_2_4_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c3">D2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E99">A167</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E96">A164</font></span></a></span>;<br/>
<a NAME="E15:61_1_2_1_1_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A171</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c3">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="lab"><font color="Green" title="E104">A172</font></span>: <font color="Maroon" title="c17">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A143</font></span></a>, <a class="txt" href="#E34:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A145</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E95">A163</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E97">A165</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c4">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="c3">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c3">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E103">A171</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
;<br/>
<a NAME="E17:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E105">A173</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E99">A167</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E96">A164</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E18:61_1_2_1_1_2_4_2"/><span class="lab"><font color="Green" title="E106">A174</font></span>: 
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_4_2_3"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_4_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2_4_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_2_1_1_2_4_2_3_1_1"/>
( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c24">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="61_1_2_1_1_2_4_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_2_1_1_2_4_2_3_1_1_1"/><span class="lab"><font color="Green" title="E107">A175</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )</span><br/><div class="add"><a NAME="E2:61_1_2_1_1_2_4_2_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A176</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E99">A167</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E96">A164</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:61_1_2_1_1_2_4_2_3_1_1_1"/><span class="lab"><font color="Green" title="E109">A177</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c3">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E99">A167</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E96">A164</font></span></a>, <a class="txt" href="#E13:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E102">A170</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E107">A175</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E110">A178</font></span>:  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,1)</span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E107">A175</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A131</font></span></a>, <a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E108">A176</font></span></a>, <a class="txt" href="#E3:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E109">A177</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/><a NAME="E5:61_1_2_1_1_2_4_2_3_1_1_1"/><span class="lab"><font color="Green" title="E111">A179</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A148</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:61_1_2_1_1_2_4_2_3_1_1_1"/><span class="lab"><font color="Green" title="E112">A180</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 ;<br/><span class="lab"><font color="Green" title="E113">A181</font></span>:  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,1)</span>)</span> = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E107">A175</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="txt" href="#E6:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E112">A180</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/><span class="lab"><font color="Green" title="E114">A182</font></span>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E107">A175</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E89">A157</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E111">A179</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="c3">D2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E89">A157</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E111">A179</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="ref" href="integra1.html#D19" target="_self" title="INTEGRA1:def.19">Def21</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:61_1_2_1_1_2_4_2_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E107">A175</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E113">A181</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E110">A178</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )</span><br/><a NAME="E10:61_1_2_1_1_2_4_2_3_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E107">A175</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E113">A181</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E110">A178</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E11:61_1_2_1_1_2_4_2_3_1_1_1"/>
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E8:61_1_2_1_1_2_4_2_3_1_1_1"><span class="lab"><font color="Green" title="E114">A182</font></span></a>, <a class="ref" href="integra1.html#T8" target="_self" title="INTEGRA1:th.8">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2_1_1_2_4_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="lab"><font color="Green" title="E107">A183</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )</span><br/><div class="add"><a NAME="E2:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="lab"><font color="Green" title="E108">A184</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="61_1_2_1_1_2_4_2_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_2_1_1_2_4_2_3_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:61_1_2_1_1_2_4_2_3_1_1_2_1"/><span class="kw">then </span>
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="c24">n</font>
 
;<br/>
<a NAME="E3:61_1_2_1_1_2_4_2_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E80">A148</font></span></a>, <a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E4:61_1_2_1_1_2_4_2_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:61_1_2_1_1_2_4_2_3_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E92">A160</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E109">A185</font></span>:  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>
<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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E103">A171</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E108">A184</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E103">A171</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E108">A184</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
;<br/><a NAME="E4:61_1_2_1_1_2_4_2_3_1_1_2"/>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">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="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E6:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A186</font></span>: 
<font color="Maroon" title="c24">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="c23">MD2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="txt" href="#E10:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E100">A168</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E98">A166</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="lab"><font color="Green" title="E111">A187</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E70">A135</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:61_1_2_1_1_2_4_2_3_1_1_2"/>
 not <font color="Maroon" title="c24">n</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E92">A160</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E107">A183</font></span></a>, <a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></span>;<br/><a NAME="E9:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T29" title="NAT_2:th.29">NAT_2:29</a></span>;<br/><a NAME="E10:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A188</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">n1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E12:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="lab"><font color="Green" title="E113">A189</font></span>: 
<font color="Maroon" title="c24">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="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E92">A160</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">n1</font> = <font color="Maroon" title="c25">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E14:61_1_2_1_1_2_4_2_3_1_1_2"/><span class="lab"><font color="Green" title="E114">A190</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E105">A173</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E113">A189</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><span class="lab"><font color="Green" title="E115">A191</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c24">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font>

;<br/><span class="lab"><font color="Green" title="E116">A192</font></span>:  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> = 
<font color="Maroon" title="c23">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E107">A183</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A131</font></span></a>, <a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E88">A156</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E82">A150</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E111">A187</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E113">A189</font></span></a>, <a class="txt" href="#E14:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E114">A190</font></span></a>, <a class="txt" href="#E10:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E112">A188</font></span></a>, <a class="txt" href="#E6:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E110">A186</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/><span class="lab"><font color="Green" title="E117">A193</font></span>:  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> = 
<font color="Maroon" title="c23">MD2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E107">A183</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A131</font></span></a>, <a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="txt" href="#E26:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E88">A156</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E82">A150</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E92">A160</font></span></a>, <a class="txt" href="#E3:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E94">A162</font></span></a>, <a class="txt" href="#E10:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E100">A168</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E98">A166</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E111">A187</font></span></a>, <a class="txt" href="#E15:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E115">A191</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/><span class="kw">hence </span><a NAME="E18:61_1_2_1_1_2_4_2_3_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E116">A192</font></span></a>, <a class="txt" href="#E3:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E109">A185</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )</span><br/><a NAME="E19:61_1_2_1_1_2_4_2_3_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E116">A192</font></span></a>, <a class="txt" href="#E17:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E117">A193</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E20:61_1_2_1_1_2_4_2_3_1_1_2"/>
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E3:61_1_2_1_1_2_4_2_3_1_1_2"><span class="lab"><font color="Green" title="E109">A185</font></span></a>, <a class="ref" href="integra1.html#T8" target="_self" title="INTEGRA1:th.8">Th10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:61_1_2_1_1_2_4_2_3"/>
(  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c23">MD2</font>,<font color="Maroon" title="c24">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="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>) &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c3">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">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="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:61_1_2_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">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="c23">MD2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c3">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:61_1_2_1_1_2_4_2"/>
<font color="Maroon" title="c17">q2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">MD2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E101">A169</font></span></a>, <a class="txt" href="#E16:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E104">A172</font></span></a>, <a class="txt" href="#E18:61_1_2_1_1_2_4_2"><span class="lab"><font color="Green" title="E106">A174</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:61_1_2_1_1_2_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 
;<br/>
<a NAME="E21:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A194</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E22:61_1_2_1_1_2_4"/>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">MD1</font> =  <a href="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) as    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E8:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A114</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E68">A133</font></span></a>, <a class="txt" href="#E21:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E92">A194</font></span></a>, <a class="ref" href="integra1.html#T37" target="_self" title="INTEGRA1:th.37">Th39</a>, <a class="ref" href="seq_4.html#T137" title="SEQ_4:th.137">SEQ_4:137</a></span>;<br/>
<a NAME="E24:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E93">A195</font></span>: 
<font color="Maroon" title="c22">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1_2_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c25">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> <span class="kw"> such that </span><br/><a NAME="E2:61_1_2_1_1_2_4_3"/><span class="lab"><font color="Green" title="E94">A196</font></span>: 
 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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> holds <br/><font color="Maroon" title="c25">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="rfunct_1.html#T71" title="RFUNCT_1:th.71">RFUNCT_1:71</a></span>;<br/>
<a NAME="E3:61_1_2_1_1_2_4_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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">g</font></span>)</span> holds <br/><font color="Maroon" title="c25">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">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="61_1_2_1_1_2_4_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">g</font></span>)</span> implies <font color="Maroon" title="c25">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font> )</span><br/>

<a NAME="E1:61_1_2_1_1_2_4_3_1"/><span class="lab"><font color="Green" title="E95">A197</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:61_1_2_1_1_2_4_3_1"/>
<font color="Maroon" title="c26">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">g</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font></span><br/>

<a NAME="E3:61_1_2_1_1_2_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A198</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:61_1_2_1_1_2_4_3_1"/><span class="lab"><font color="Green" title="E97">A199</font></span>: 
<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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">x</font> = <font color="Maroon" title="c26">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_1_1_2_4_3_1"><span class="lab"><font color="Green" title="E96">A198</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_3_1"><span class="lab"><font color="Green" title="E95">A197</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:61_1_2_1_1_2_4_3_1"/>
<font color="Maroon" title="c25">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2_1_1_2_4_3"><span class="lab"><font color="Green" title="E94">A196</font></span></a>, <a class="txt" href="#E3:61_1_2_1_1_2_4_3_1"><span class="lab"><font color="Green" title="E96">A198</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_4_3_1"><span class="lab"><font color="Green" title="E97">A199</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4_3_1"><span class="lab"><font color="Green" title="E95">A197</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:61_1_2_1_1_2_4_3_1"/>
<font color="Maroon" title="c25">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">g</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_1_1_2_4_3_1"><span class="lab"><font color="Green" title="E96">A198</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:61_1_2_1_1_2_4_3"/>
<font color="Maroon" title="c22">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="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> is  <a href="seq_2.html#V2" title="SEQ_2:attr.2">bounded_below</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T71" title="RFUNCT_1:th.71">RFUNCT_1:71</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:61_1_2_1_1_2_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2"><span class="lab"><font color="Green" title="E51">A106</font></span></a>, <a class="txt" href="#E5:61_1_2"><span class="lab"><font color="Green" title="E52">A107</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E26:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A200</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E27:61_1_2_1_1_2_4"/><span class="lab"><font color="Green" title="E95">A201</font></span>: 
 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="c13">q1</font> holds <br/><font color="Maroon" title="c13">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">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="61_1_2_1_1_2_4_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:61_1_2_1_1_2_4_4"/>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A110</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/>
<a NAME="E2:61_1_2_1_1_2_4_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A202</font></span>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<span class="lab"><font color="Green" title="E97">A203</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E96">A202</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<a NAME="E5:61_1_2_1_1_2_4_4"/><span class="lab"><font color="Green" title="E98">A204</font></span>: 
<font color="Maroon" title="c24">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="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2"><span class="lab"><font color="Green" title="E51">A106</font></span></a>, <a class="txt" href="#E5:61_1_2"><span class="lab"><font color="Green" title="E52">A107</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E6:61_1_2_1_1_2_4_4"/>
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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A205</font></span>: 
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="c24">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E94">A200</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A206</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c24">MD1</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">MD1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c25">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n</font> &amp; <font color="Maroon" title="c25">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="c13">q1</font> implies <font color="Maroon" title="c13">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:61_1_2_1_1_2_4_4"/><span class="lab"><font color="Green" title="E101">A207</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n</font>
 <span class="kw">and </span><br/><a NAME="E10:61_1_2_1_1_2_4_4"/><span class="lab"><font color="Green" title="E102">A208</font></span>: 
<font color="Maroon" title="c25">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="c13">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font></span><br/>

<a NAME="E11:61_1_2_1_1_2_4_4"/><span class="lab"><font color="Green" title="E103">A209</font></span>: 
<font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A112</font></span></a>, <a class="txt" href="#E9:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E101">A207</font></span></a>, <a class="txt" href="#E10:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E102">A208</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E12:61_1_2_1_1_2_4_4"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c24">MD1</font>,1)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E99">A205</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E13:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A210</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c24">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="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E100">A206</font></span></a>, <a class="txt" href="#E5:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E98">A204</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E14:61_1_2_1_1_2_4_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 
;<br/>
<a NAME="E15:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A211</font></span>: 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E16:61_1_2_1_1_2_4_4"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E53">A108</font></span></a>, <a class="ref" href="integra1.html#D4" target="_self" title="INTEGRA1:def.4">Def5</a></span>;<br/>
<a NAME="E17:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A212</font></span>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><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="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E105">A211</font></span></a>, <a class="ref" href="integra1.html#T4" target="_self" title="INTEGRA1:th.4">Th5</a></span>;<br/>
<a NAME="E18:61_1_2_1_1_2_4_4"/><span class="lab"><font color="Green" title="E107">A213</font></span>: 
<font color="Maroon" title="c25">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="c13">q1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E101">A207</font></span></a>, <a class="txt" href="#E10:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E102">A208</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E19:61_1_2_1_1_2_4_4"/><span class="kw">then </span>
<font color="Maroon" title="c25">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="c24">MD1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A112</font></span></a>, <a class="txt" href="#E26:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E94">A200</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E20:61_1_2_1_1_2_4_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A214</font></span>: 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c22">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="c2">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#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="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E103">A209</font></span></a>, <a class="txt" href="#E13:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E104">A210</font></span></a>, <a class="txt" href="#E15:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E105">A211</font></span></a>, <a class="txt" href="#E17:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E106">A212</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E21:61_1_2_1_1_2_4_4"/>
<font color="Maroon" title="c25">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E107">A213</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c13">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A111</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E58">A113</font></span></a>, <a class="txt" href="#E11:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E103">A209</font></span></a>, <a class="txt" href="#E4:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E97">A203</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</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="c4">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="c2">D1</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#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="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_2"><span class="lab"><font color="Green" title="E49">A104</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E23:61_1_2_1_1_2_4_4"/>
<font color="Maroon" title="c13">q1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">MD1</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c25">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:61_1_2_1_1_2_4_4"><span class="lab"><font color="Green" title="E108">A214</font></span></a>, <a class="ref" href="funct_1.html#T51" title="FUNCT_1:th.51">FUNCT_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E28:61_1_2_1_1_2_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">MD1</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A112</font></span></a>, <a class="txt" href="#E26:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E94">A200</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E29:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A215</font></span>: 
<font color="Maroon" title="c13">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">MD1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E95">A201</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<a NAME="E30:61_1_2_1_1_2_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2"><span class="lab"><font color="Green" title="E51">A106</font></span></a>, <a class="txt" href="#E5:61_1_2"><span class="lab"><font color="Green" title="E52">A107</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>;<br/>
<a NAME="E31:61_1_2_1_1_2_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">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="c5">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> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E32:61_1_2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A216</font></span>: 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">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="c22">g</font>,<font color="Maroon" title="c23">MD2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E93">A195</font></span></a>, <a class="txt" href="#E8:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="ref" href="integra1.html#T31" target="_self" title="INTEGRA1:th.31">Th33</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c23">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="finseq_6.html#K3" title="FINSEQ_6:func.3">mid</a> (<font color="Maroon" title="c3">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A131</font></span></a>, <a class="txt" href="#E35:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E78">A146</font></span></a>, <a class="txt" href="#E11:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E87">A155</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E88">A156</font></span></a>, <a class="txt" href="#E1:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E82">A150</font></span></a>, <a class="txt" href="#E18:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E90">A158</font></span></a>, <a class="ref" href="finseq_6.html#T118" title="FINSEQ_6:th.118">FINSEQ_6:118</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E34:61_1_2_1_1_2_4"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">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="c17">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E76">A144</font></span></a>, <a class="txt" href="#E29:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E96">A215</font></span></a>, <a class="txt" href="#E19:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E91">A159</font></span></a>, <a class="txt" href="#E32:61_1_2_1_1_2_4"><span class="lab"><font color="Green" title="E97">A216</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c18">KD2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>;<br/><a NAME="E40:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E82">A217</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c16">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="c17">q2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A145</font></span></a>, <a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>;<br/><a NAME="E41:61_1_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A218</font></span>: 
 <a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E73">A141</font></span></a>, <a class="txt" href="#E37:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A147</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E42:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A219</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E75">A143</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/><a NAME="E43:61_1_2_1_1_2"/>
 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">p2</font> holds <br/><font color="Maroon" title="c16">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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="61_1_2_1_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c19">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">p2</font> implies <font color="Maroon" title="c16">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:61_1_2_1_1_2_5"/><span class="lab"><font color="Green" title="E85">A220</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">and </span><br/><a NAME="E2:61_1_2_1_1_2_5"/><span class="lab"><font color="Green" title="E86">A221</font></span>: 
<font color="Maroon" title="c19">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">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font></span><br/>

<a NAME="E3:61_1_2_1_1_2_5"/><span class="lab"><font color="Green" title="E87">A222</font></span>: 
<font color="Maroon" title="c19">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">p2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E85">A220</font></span></a>, <a class="txt" href="#E2:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E86">A221</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:61_1_2_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A223</font></span>: 
<font color="Maroon" title="c19">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A219</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:61_1_2_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A224</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<span class="lab"><font color="Green" title="E90">A225</font></span>:  <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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E74">A142</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>
;<br/>
<a NAME="E7:61_1_2_1_1_2_5"/><span class="lab"><font color="Green" title="E91">A226</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A138</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A218</font></span></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>
;<br/>
<a NAME="E9:61_1_2_1_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c19">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E88">A223</font></span></a>, <a class="txt" href="#E7:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E91">A226</font></span></a>, <a class="txt" href="#E6:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E90">A225</font></span></a></span>;<br/>
<a NAME="E10:61_1_2_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A227</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>;<br/>
<span class="lab"><font color="Green" title="E93">A228</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E92">A227</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<span class="lab"><font color="Green" title="E94">A229</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D15" title="FINSEQ_1:def.15">FINSEQ_1:def 15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">D2</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E89">A224</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
;<br/>
<a NAME="E13:61_1_2_1_1_2_5"/>
<font color="Maroon" title="c19">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">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E87">A222</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:61_1_2_1_1_2_5"/>
<font color="Maroon" title="c16">p2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E77">A145</font></span></a>, <a class="txt" href="#E12:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E94">A229</font></span></a>, <a class="txt" href="#E11:61_1_2_1_1_2_5"><span class="lab"><font color="Green" title="E93">A228</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E44:61_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A230</font></span>: 
<font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c5">k</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A219</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/><a NAME="E45:61_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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c12">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="c13">q1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E58">A113</font></span></a>, <a class="ref" href="rvsum_1.html#T75" title="RVSUM_1:th.75">RVSUM_1:75</a></span>;<br/><a NAME="E46:61_1_2_1_1_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c13">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c12">p1</font></span>)</span>
 ;<br/><a NAME="E47:61_1_2_1_1_2"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c17">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="c12">p1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E81">A149</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E48:61_1_2_1_1_2"/><span class="kw">then </span>
<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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c17">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="c12">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E49:61_1_2_1_1_2"/><span class="kw">then </span>
<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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c17">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">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_2"><span class="lab"><font color="Green" title="E48">A103</font></span></a>, <a class="txt" href="#E20:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E67">A132</font></span></a>, <a class="txt" href="#E13:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E63">A128</font></span></a>, <a class="txt" href="#E44:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A230</font></span></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></span>;<br/><span class="kw">hence </span><a NAME="E50:61_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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:61_1_2_1_1_2"><span class="lab"><font color="Green" title="E82">A217</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:61_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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:61_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="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E2:61_1"><span class="lab"><font color="Green" title="E47">A102</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:61"/>
 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="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c2">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#K5" title="INTEGRA1:func.5">lower_volume</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">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#K13" title="INTEGRA1:func.13">indx</a> (<font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c2">D1</font>,<font color="Olive" title="b1">i</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
