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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">T</font> be   <a href="integra2.html#NM1">DivSequence</a> of <font color="Maroon">A</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E39</font></i>: 
 <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="fdiff_1.html#V1">convergent_to_0</a>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E40</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E4:32"/><i><font color="Green">E41</font></i>: 
(  <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a>  &amp;  <a href="integra2.html#K2">delta</a> <font color="Maroon">T</font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="integra3.html#E3"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="fdiff_1.html#D1">FDIFF_1:def 1</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">e</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>
<a NAME="E4:32_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

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

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

<a NAME="E2:32_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:32_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E4:32_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E5:32_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="integra3.html#T1" target="_self">Th1</a>, <a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/>
<a NAME="E6:32_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E7:32_1_1"/>
 <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D19">INTEGRA1:def 19</a></i>;<br/>
<a NAME="E8:32_1_1"/><b>then </b>
 <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:32_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> &amp;  <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E11:32_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">D</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E13:32_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font>
 ;<br/>
<a NAME="E14:32_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="integra3.html#E4:32_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E15:32_1_1"/><b>then </b>
 <a href="integra1.html#K16">delta</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:32_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="integra3.html#E4:32_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<b>hence </b><a NAME="E16:32_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32"><i><font color="Green">E40</font></i></a>, <a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a>, <a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_1"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:32"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">D</font>, <font color="Olive">D1</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>  ex <font color="Olive">D2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> st <br/>( <font color="Olive">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp; <font color="Olive">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Olive">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Olive">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>, <font color="Maroon">D1</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>

<b>consider </b><font color="Maroon">D2</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:32_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E3:32_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E4:32_2"/><b>then </b>
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="integra1.html#T48">INTEGRA1:48</a></i>;<br/>
<a NAME="E5:32_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E6:32_2"/>
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green">E39</font></i></a>, <a class="txt" href="integra3.html#E3:32"><i><font color="Green">E40</font></i></a>, <a class="ref" href="integra1.html#T48">INTEGRA1:48</a></i>;<br/>
<a NAME="E7:32_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2"/>
 ex <font color="Olive">D2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green">E39</font></i></a>, <a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32"/><i><font color="Green">E58</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">D</font>, <font color="Olive">D1</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>  st  <a href="integra1.html#K16">delta</a> <font color="Olive">D1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Olive">D</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">D2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> st <br/>( <font color="Olive">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp; <font color="Olive">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Olive">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Olive">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Olive">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>, <font color="Maroon">D1</font> be    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:32_3"/><i><font color="Green">E60</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_3"/>
 ex <font color="Olive">D2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">D2</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:32_3_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E3:32_3_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>,  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="integra1.html#K18">PartSums</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>) <b>-&gt; </b>   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:32_3_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Olive">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Olive">j</font> holds <br/><font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:32_3_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<b>assume </b><a NAME="E2:32_3_1_1_1_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 ;<br/>

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

<a NAME="E4:32_3_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E5:32_3_1_1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_1"/><i><font color="Green">E72</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_1"/>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_1_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 )
 <b>by </b><i><a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><a NAME="E11:32_3_1_1_1_1_1_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_1_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> 1
 <b>by </b><i>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_1_1_1_1_1_1"/>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_1_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T2" target="_self">Th2</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E4:32_3_1_1_1_1_1_1_2"/>
<font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra3.html#T2" target="_self">Th2</a>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E6:32_3_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E7:32_3_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><a NAME="E9:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><a NAME="E10:32_3_1_1_1_1_1_1_2"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E12:32_3_1_1_1_1_1_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:32_3_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E14:32_3_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:32_3_1_1_1_1_1_1_2"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E16:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E17:32_3_1_1_1_1_1_1_2"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/><b>hence </b><a NAME="E18:32_3_1_1_1_1_1_1_2"/>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E12:32_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> implies  ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Olive">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> ) );<br/>
<a NAME="E3:32_3_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T8">INTEGRA1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:32_3_1_1_1_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:32_3_1_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E106</font></i>: 
( <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E107</font></i>: 
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1"/><i><font color="Green">E108</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_2_1_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>

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


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1_1_2_1"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,1,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> is <a href="goboard1.html#V1">increasing</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T37">INTEGRA1:37</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> is <a href="goboard1.html#V1">increasing</a>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_2_1_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_2_1_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:32_3_1_1_1_2_1"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,1,<font color="Maroon">j1</font> is <a href="goboard1.html#V1">increasing</a>
 
<b>by </b><i><a class="ref" href="integra1.html#T37">INTEGRA1:37</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font> is <a href="goboard1.html#V1">increasing</a>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1"/><i><font color="Green">E112</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_3"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green">E115</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_3_1"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_3_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_3_1"/><i><font color="Green">E119</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_3_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> implies <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_1_1"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_1_1"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_2_1_3_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1_3_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> implies <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_2_1_3_1_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_3_1_2"/><b>then </b>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1_2"/><b>then </b><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1_2"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_2_1_3_1_2"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_3_1_1_1_2_1_3_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> implies <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_1_3"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_1_3"/><b>then </b><i><font color="Green">E126</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_1_3"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_3_1_3"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1_2_1_3_1_3"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:32_3_1_1_1_2_1_3_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , , <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_3"/><b>then </b><i><font color="Green">E128</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_3"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_3_2"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green">E131</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:32_3_1_1_1_2_1_3_2"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_3_2"/><i><font color="Green">E133</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1_3_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_2_1_3_2"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_2_1_3_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E22:32_3_1_1_1_2_1_3_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_2_1_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_2_1_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> holds <br/><font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_4"/><i><font color="Green">E141</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> )
 ;<br/>

<b>assume </b><a NAME="E2:32_3_1_1_1_2_1_4"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_4_1_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_4_1_1_1"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_4_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_4_1_1_1"/><i><font color="Green">E148</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E11:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:32_3_1_1_1_2_1_4_1_1_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E13:32_3_1_1_1_2_1_4_1_1_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E14:32_3_1_1_1_2_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_4_1_1_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_4_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E153</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/><a NAME="E7:32_3_1_1_1_2_1_4_1_1_2"/>
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, </i>;<br/><a NAME="E8:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E154</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:32_3_1_1_1_2_1_4_1_1_2"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E12:32_3_1_1_1_2_1_4_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E13:32_3_1_1_1_2_1_4_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:32_3_1_1_1_2_1"/><i><font color="Green">E156</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1"/>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E158</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_2_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E22:32_3_1_1_1_2_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E23:32_3_1_1_1_2_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E24:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E25:32_3_1_1_1_2_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_5"/><i><font color="Green">E160</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_2_1_5"/><i><font color="Green">E161</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green">E162</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_5"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_5"/><i><font color="Green">E165</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_5_1"/><i><font color="Green">E166</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_5_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_5_1"/><b>then </b><i><font color="Green">E168</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_5_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_5_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_5_1_1_1_1"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_5_1_1_1_1"/><b>then </b><i><font color="Green">E170</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_5_1_1_1_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, </i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_2_1_5_1_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_5_1_1_1_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green">E172</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_5_1_1_1_2"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b>
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E7:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, </i>;<br/><a NAME="E11:32_3_1_1_1_2_1_5_1_1_1_2"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, , </i>;<br/><a NAME="E12:32_3_1_1_1_2_1_5_1_1_1_2"/><b>then </b><i><font color="Green">E176</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E13:32_3_1_1_1_2_1_5_1_1_1_2"/>
<font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E14:32_3_1_1_1_2_1_5_1_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , , , , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_3_1_1_1_2_1_5_1"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:32_3_1_1_1_2_1_5"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_5"/><b>then </b>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E177</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E11:32_3_1_1_1_2_1_5"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="integra3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E13:32_3_1_1_1_2_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green">E179</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1_5"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1_5"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1_5"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E20:32_3_1_1_1_2_1_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E27:32_3_1_1_1_2_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E28:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E182</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E29:32_3_1_1_1_2_1"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:32_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<i><font color="Green">E184</font></i>: <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j1</font>)
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>
;<br/>
<a NAME="E32:32_3_1_1_1_2_1"/><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_7"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_7"/><b>then </b>
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_7"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_7"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E33:32_3_1_1_1_2_1"/><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8"/><i><font color="Green">E187</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green">E188</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8"/><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/>

<b>reconsider </b><font color="Maroon">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">ID2</font> = <font color="Maroon">ID1</font> <a href="nat_1.html#K1">+</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E191</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ID1</font> &amp; <font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ID2</font> &amp; <font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_2_1_8_2_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ID1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<b>thus </b><a NAME="E2:32_3_1_1_1_2_1_8_2_1"/>
<font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">ID2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<a NAME="E3:32_3_1_1_1_2_1_8_2_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_1"/>
<font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E192</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_2"/><b>then </b><i><font color="Green">E194</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ID1</font> &amp; <font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ID1</font> &amp; <font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_2_2"/><b>then </b>
<font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_2"/>
<font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_3"/>
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ID2</font> &amp; <font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">ID2</font> &amp; <font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_2_3"/><b>then </b>
<font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_3"/>
<font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_2_1_8_2"/><b>then </b><i><font color="Green">E197</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_4"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_4"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_2_4"/>
( not <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> &amp; not <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> or <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> or <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> )
 </span><b>by </b><i><a class="ref" href="integra3.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_8_2_4_1_1_1"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_8_2_4_1_1_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_2_4_1_1_1"/><b>then </b>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_8_2_4_1_1_1"/><b>then </b>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E7:32_3_1_1_1_2_1_8_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_2_4_1_1_2"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_2_1_8_2_4_1_1_2"/><i><font color="Green">E203</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_2_1_8_2_4_1_1_2"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_2_4_1_1_2"/><b>then </b>
( <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_8_2_4_1_1_2"/><b>then </b>
( <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E7:32_3_1_1_1_2_1_8_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_2_1_8_2_4"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E204</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_5"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_2_5"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:32_3_1_1_1_2_1_8_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_6"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_6"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_2_6"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1_8_2"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_2_7"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_2_7"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_2_7"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_3_1_1_1_2_1_8_2"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">ID2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="integra3.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>hence </b><a NAME="E15:32_3_1_1_1_2_1_8_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="goboard2.html#T19">GOBOARD2:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_2_1_8"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_3"/><i><font color="Green">E235</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/>

<b>thus </b><a NAME="E4:32_3_1_1_1_2_1_8_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green">E236</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8"/><i><font color="Green">E237</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_4"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1_1_2_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green">E238</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_8"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_8"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_8"/><b>then </b><i><font color="Green">E239</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_8"/><i><font color="Green">E240</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:32_3_1_1_1_2_1_8"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_2_1_8_6_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green">E242</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green">E243</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_1_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_2_1_8_6_1_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green">E244</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_1_2_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_6_1_1_2_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_1_2"/><b>then </b><i><font color="Green">E245</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_1_2"/><i><font color="Green">E246</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_1_2"/><i><font color="Green">E247</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_8_6_1_1_2"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_6_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra3.html#T7" target="_self">Th7</a>, <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_2_1_8_6_1_1_2"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:32_3_1_1_1_2_1_8_6_1_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_6_1_1"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then </b><i><font color="Green">E248</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/><a NAME="E7:32_3_1_1_1_2_1_8_6_1_1"/><i><font color="Green">E249</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E8:32_3_1_1_1_2_1_8_6_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_8_6_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_1_1_2_1_8_6_1_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E250</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E251</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_1"/><i><font color="Green">E252</font></i>: 
2 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_1"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E4:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b><i><font color="Green">E253</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/><a NAME="E6:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E254</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2"/><i><font color="Green">E255</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2"/><i><font color="Green">E256</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_2_2"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_2"/><i><font color="Green">E257</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green">E258</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_2_2_3"/><b>then </b><i><font color="Green">E259</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green">E260</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green">E261</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green">E262</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E8:32_3_1_1_1_2_1_8_6_1_2_2_3"/><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 
;<br/>
<b>thus </b><a NAME="E9:32_3_1_1_1_2_1_8_6_1_2_2_3"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i>, , <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E10:32_3_1_1_1_2_1_8_6_1_2_2_3"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, , <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_2_1_8_6_1_2_2"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T7" target="_self">Th7</a>, <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<b>then </b> <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_2_1_8_6_1_2_2"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E8:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E265</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E266</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/><a NAME="E10:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E267</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_3"/><i><font color="Green">E268</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_3"/><i><font color="Green">E269</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_8_6_1_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_8_6_1_2_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_8_6_1_2_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_3_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green">E270</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">ID1</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">ID2</font> = <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2;<br/>
<i><font color="Green">E271</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra3.html#T5" target="_self">Th5</a>, , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E5:32_3_1_1_1_2_1_8_6_1_2_3"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, , <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green">E272</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E10:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green">E273</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_2_1_8_6_1_2_3"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E13:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green">E274</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b><i><font color="Green">E275</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>;<br/>
<a NAME="E20:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E22:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E23:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E24:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E25:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E26:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E27:32_3_1_1_1_2_1_8_6_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E28:32_3_1_1_1_2_1_8_6_1_2_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E11:32_3_1_1_1_2_1_8_6_1_2"/><i><font color="Green">E276</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:32_3_1_1_1_2_1_8_6_1_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="integra3.html#E2:32"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E13:32_3_1_1_1_2_1_8_6_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E14:32_3_1_1_1_2_1_8_6_1_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T6" target="_self">Th6</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:32_3_1_1_1_2_1_8"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E34:32_3_1_1_1_2_1"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_9"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_9"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_9"/><b>then </b><i><font color="Green">E278</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_9"/><b>then </b><i><font color="Green">E279</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_9"/><b>then </b><i><font color="Green">E280</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) )
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_9"/><i><font color="Green">E281</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)</span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_9"/><i><font color="Green">E282</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_9"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_9"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_9"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="integra2.html#T4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E11:32_3_1_1_1_2_1_9"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>)
 <b>by </b><i>, <a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:32_3_1_1_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_2_1_10"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_2_1_10"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_2_1_10"/><b>then </b><i><font color="Green">E284</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_2_1_10"/><b>then </b><i><font color="Green">E285</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) )
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_2_1_10"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)</span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_2_1_10"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_2_1_10"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_2_1_10"/>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_2_1_10"/><b>then </b>
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_2_1_10"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="finseq_1.html#K17">|</a> <font color="Maroon">j1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),1,<font color="Maroon">j1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),1,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , , <a class="ref" href="integra2.html#T4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="finseq_1.html#K17">|</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E11:32_3_1_1_1_2_1_10"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p3">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>)
 <b>by </b><i>, <a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:32_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p4">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <span class="p4">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E37:32_3_1_1_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:32_3_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1"/><i><font color="Green">E288</font></i>: 
for <font color="Olive">i</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<a NAME="E2:32_3_1_1_1_3"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#T8">INTEGRA1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:32_3_1_1_1_3_1"/><i><font color="Green">E292</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:32_3_1_1_1_3_1"/><i><font color="Green">E293</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E294</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E295</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:32_3_1_1_1_3_1"/><i><font color="Green">E297</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_3_1"/><i><font color="Green">E298</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E13:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E299</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:32_3_1_1_1_3_1"/><i><font color="Green">E300</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_1"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_1_1_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E302</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E303</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a>, , <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, </i>;<br/><a NAME="E7:32_3_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="goboard2.html#T19">GOBOARD2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E304</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E305</font></i>: 
<font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_1_1_1_2"/><i><font color="Green">E306</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_1_1_1_2"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E308</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E309</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_1_1_1_2"/><i><font color="Green">E310</font></i>: 
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_1_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_1_1_1_2_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_1_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E12:32_3_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_1_1_1_2"/><b>then </b><i><font color="Green">E311</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:32_3_1_1_1_3_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><b>hence </b><a NAME="E16:32_3_1_1_1_3_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E312</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E313</font></i>: 
<font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_3_1"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_3_1"/><i><font color="Green">E315</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E316</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E317</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E318</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E22:32_3_1_1_1_3_1"/><i><font color="Green">E319</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E23:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E24:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E320</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E25:32_3_1_1_1_3_1"/><i><font color="Green">E321</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:32_3_1_1_1_3_1"/><i><font color="Green">E322</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp;  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E27:32_3_1_1_1_3_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E28:32_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E29:32_3_1_1_1_3_1"/><b>then </b><i><font color="Green">E323</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E30:32_3_1_1_1_3_1"/>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E31:32_3_1_1_1_3_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>then </b><i><font color="Green">E324</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>
;<br/>
<a NAME="E33:32_3_1_1_1_3_1"/><i><font color="Green">E325</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1"/>
( <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E326</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E327</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E328</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><i><font color="Green">E329</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
1

;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p5">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p0">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p5">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E330</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p5">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E331</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E332</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E333</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_3"/>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_3"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E334</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E335</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E336</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E19:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E337</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><b>then </b><i><font color="Green">E338</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="integra3.html#T7" target="_self">Th7</a>, <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span>

;<br/><a NAME="E21:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E339</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5"/><b>then </b><i><font color="Green">E340</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E341</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E342</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E343</font></i>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E344</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_2"/><i><font color="Green">E345</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_2"/>
not <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/><i><font color="Green">E346</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/><b>then </b>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , , <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_5_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_2"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E347</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_3"/><i><font color="Green">E348</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_3"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_3"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_3"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E349</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_4"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_1_5"/><i><font color="Green">E350</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_5"/><i><font color="Green">E351</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_5"/>
not <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/><i><font color="Green">E352</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/><b>then </b>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, , , <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_5_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_5"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_1_5"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_5_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_5_6"/><i><font color="Green">E353</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_5_6"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_5_6"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_5_6"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_5"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E12:32_3_1_1_1_3_1_3_1_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:32_3_1_1_1_3_1_3_1_1_1"/><i><font color="Green">E354</font></i>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_6"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_6"/><b>then </b><i><font color="Green">E355</font></i>: 
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> or <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> implies <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_6_1"/><b>then </b><i><font color="Green">E356</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_6_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E23:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1"/>
( <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 or <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><i><font color="Green">E357</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><b>then </b>
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><b>then </b><i><font color="Green">E358</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><b>then </b><i><font color="Green">E359</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><i><font color="Green">E360</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/><i><font color="Green">E361</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/><b>then </b><i><font color="Green">E362</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , , , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_1_2"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><i><font color="Green">E363</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/><b>then </b><i><font color="Green">E364</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i>, , <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_7_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finsop_1.html#T12">FINSOP_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E365</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E366</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E367</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E368</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E369</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b><i><font color="Green">E370</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E371</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><b>then </b><i><font color="Green">E372</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 1 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> 2 = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>

;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b>
 <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b><i><font color="Green">E373</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T107">RVSUM_1:107</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><i><font color="Green">E374</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/><b>then </b><i><font color="Green">E375</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/><i><font color="Green">E376</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, , <a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_3"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/><b>then </b><i><font color="Green">E377</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/><i><font color="Green">E378</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="txt" href="integra3.html#E1"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="integra1.html#T11">INTEGRA1:11</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/><b>then </b><i><font color="Green">E379</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b><i><font color="Green">E380</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b><i><font color="Green">E381</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><i><font color="Green">E382</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><i><font color="Green">E383</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/><b>then </b><i><font color="Green">E384</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="integra3.html#E3:32_3_1_1_1"><i><font color="Green">E102</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E11:32_3_1_1_1_3_1_3_1_1_1_7_1_2_5"/>
<span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E17:32_3_1_1_1_3_1_3_1_1_1_7_1_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E385</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E25:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/><a NAME="E26:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b><i><font color="Green">E386</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E27:32_3_1_1_1_3_1_3_1_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_1_8"/><i><font color="Green">E387</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_1_8"/><b>then </b><i><font color="Green">E388</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_1_8"/>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_1_8"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_1_8"/><b>then </b><i><font color="Green">E389</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_1_8"/>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D19">INTEGRA1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_3_1_3_1_1_1_8"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:32_3_1_1_1_3_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E29:32_3_1_1_1_3_1_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E390</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E391</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E392</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">B</font> being  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">MD1</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">B</font>, <font color="Maroon">MD2</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E393</font></i>: 
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> &amp; <font color="Maroon">MD1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">MD2</font> &amp; <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font> &amp; <font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , </i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E394</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E395</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font>
 ;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E396</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><i><font color="Green">E397</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">j</font>

;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E398</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_2"/><i><font color="Green">E399</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_2"/><i><font color="Green">E400</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_2"/><i><font color="Green">E401</font></i>: 
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T7" target="_self">Th7</a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_2"/>
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_2"/><b>then </b><i><font color="Green">E402</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> = 
<font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>

;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">B</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E403</font></i>: 
<font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_3"/><i><font color="Green">E404</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_3"/><i><font color="Green">E405</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_3_1"/><b>then </b><i><font color="Green">E406</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_3_1"/><b>then </b><i><font color="Green">E407</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_3_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_3_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_3_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_3"/><b>then </b><i><font color="Green">E408</font></i>: 
<font color="Maroon">g</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D10">RFUNCT_1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_3"/><i><font color="Green">E409</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_3"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_3_2"/><b>then </b><i><font color="Green">E410</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_3_2"/><b>then </b><i><font color="Green">E411</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_3_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_3_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_3_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_3"/><b>then </b>
<font color="Maroon">g</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_3"/>
<font color="Maroon">g</font> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E412</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E413</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E414</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E19:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E415</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 ;<br/><a NAME="E20:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E416</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1"/><i><font color="Green">E417</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1"/><i><font color="Green">E418</font></i>: 
<font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1"/>
<font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , <a class="ref" href="integra3.html#T8" target="_self">Th8</a>, <a class="txt" href="integra3.html#E13"><i><font color="Green">Lemma84</font></i></a>, <a class="txt" href="integra3.html#E14"><i><font color="Green">Lemma86</font></i></a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1"/><b>then </b><i><font color="Green">E419</font></i>: 
( <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, </i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1"/><i><font color="Green">E420</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T28">JORDAN3:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><i><font color="Green">E421</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><i><font color="Green">E422</font></i>: 
( <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b><i><font color="Green">E423</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font> &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font>
 ;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b><i><font color="Green">E424</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><i><font color="Green">E425</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/><b>then </b><i><font color="Green">E426</font></i>: 
<font color="Maroon">MD1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then </b><font color="Maroon">MD1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>

;<br/><b>hence </b><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><i><font color="Green">E427</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b><i><font color="Green">E429</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
not <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b><i><font color="Green">E430</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b><i><font color="Green">E431</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b>
( <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><b>then </b><i><font color="Green">E432</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><i><font color="Green">E433</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/><i><font color="Green">E435</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , , , </i>;<br/>


</div><b>end;</b></div><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_4_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4"/><b>then </b><i><font color="Green">E436</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4"/>
for <font color="Olive">x1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2"/><i><font color="Green">E437</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font> or <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E438</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T28">JORDAN3:28</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green">E439</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T34">FINSEQ_3:34</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green">E440</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E441</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E442</font></i>: 
<font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">j</font>

;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b>
<font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green">E443</font></i>: 
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, , </i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D20">INTEGRA1:def 20</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E444</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E445</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> )
 <b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green">E446</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green">E447</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E22:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E448</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E23:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><i><font color="Green">E449</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E24:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/><b>then </b><i><font color="Green">E450</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD2</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E25:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>then </b><font color="Maroon">MD2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E27:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 <b>by </b><i>, <a class="txt" href="integra3.html#E3:32_3_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b><i><font color="Green">E451</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><i><font color="Green">E452</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="integra1.html#D20">INTEGRA1:def 20</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><i><font color="Green">E453</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, , <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b><i><font color="Green">E454</font></i>: 
( <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><i><font color="Green">E455</font></i>: 
( <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">j1</font> = <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/><b>then </b>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/><b>hence </b><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_4_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 <b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">MD1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E21:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E456</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E22:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E457</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E23:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E458</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_5_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/><i><font color="Green">E459</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/><b>then </b><i><font color="Green">E460</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_5_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="integra3.html#T5" target="_self">Th5</a>, , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_5_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><i><font color="Green">E461</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b><i><font color="Green">E462</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><i><font color="Green">E463</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b><i><font color="Green">E464</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/><b>then </b><i><font color="Green">E465</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , <a class="ref" href="integra3.html#T5" target="_self">Th5</a>, <a class="txt" href="integra3.html#E15"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_5_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:32_3_1_1_1_3_1_3_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E25:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E466</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E26:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E467</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6"/>
(  <a href="integra1.html#K3">vol</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1"/><i><font color="Green">E468</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <font color="Maroon">B</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6_1"/><i><font color="Green">E469</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6_1_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_6_1"/><b>then </b><i><font color="Green">E470</font></i>: 
 <a href="integra1.html#K3">vol</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, , <a class="txt" href="integra3.html#E16"><i><font color="Green">Lemma90</font></i></a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 
<b>by </b><i>, , <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
 <a href="integra1.html#K3">vol</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/>

<a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_6_1"/><b>then </b><i><font color="Green">E471</font></i>: 
<font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_6_1"/><i><font color="Green">E472</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
<font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_6_1"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_6_1"/>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_6"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra3.html#T7" target="_self">Th7</a>, , , </i>;<br/>


</div><b>end;</b></div><a NAME="E27:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E28:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E473</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><a NAME="E29:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E474</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/><a NAME="E30:32_3_1_1_1_3_1_3_1_1_2"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/><a NAME="E31:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> is <a href="seq_4.html#V2">bounded_below</a> )
 <b>by </b><i><a class="ref" href="integra1.html#T13">INTEGRA1:13</a>, <a class="ref" href="integra1.html#T15">INTEGRA1:15</a></i>;<br/><a NAME="E32:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="seq_4.html#T64">SEQ_4:64</a>, <a class="ref" href="seq_4.html#T65">SEQ_4:65</a></i>;<br/><a NAME="E33:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E475</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E34:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i/>;<br/><a NAME="E35:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E36:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E476</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra3.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E37:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">B</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">MD1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D19">INTEGRA1:def 19</a></i>;<br/><a NAME="E38:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">B</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E40:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E477</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">B</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">MD1</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">B</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">MD1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E41:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">B</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">MD1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E42:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E478</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E43:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E479</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E44:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E45:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E46:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E480</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E47:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E481</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E48:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E482</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E49:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E50:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E51:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E483</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E52:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E484</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/><i><font color="Green">E485</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/><b>then </b><i><font color="Green">E486</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_7_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><i><font color="Green">E487</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green">E488</font></i>: 
( <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font> &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><b>then </b><i><font color="Green">E489</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font> )
 <b>by </b><i/>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32_3_1_1_1"><i><font color="Green">E102</font></i></a>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_7_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div><a NAME="E53:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E490</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E54:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E491</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/><a NAME="E55:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/><a NAME="E56:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E57:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E58:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/><a NAME="E59:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E492</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font>
 <b>by </b><i><a class="ref" href="integra1.html#D19">INTEGRA1:def 19</a></i>;<br/><a NAME="E60:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/><a NAME="E61:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E62:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E493</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">MD1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E63:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E494</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E64:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E495</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E65:32_3_1_1_1_3_1_3_1_1_2"/>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E66:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E67:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E496</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E68:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E497</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E69:32_3_1_1_1_3_1_3_1_1_2"/><b>then </b><i><font color="Green">E498</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 <b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/><a NAME="E70:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E499</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_8"/><b>then </b><i><font color="Green">E500</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_8"/><i><font color="Green">E501</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_8"/><b>then </b><i><font color="Green">E502</font></i>: 
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1"/>
( <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font> or <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> )
 </span><b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/><i><font color="Green">E504</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/><b>then </b><i><font color="Green">E505</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_8_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green">E506</font></i>: 
( <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green">E507</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green">E508</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_2"/>
<font color="Maroon">n1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b><i><font color="Green">E509</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green">E510</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green">E511</font></i>: 
( <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b><i><font color="Green">E512</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:32_3_1_1_1"><i><font color="Green">E64</font></i></a>, , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/><a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><i><font color="Green">E513</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/><a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E2:32_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_8_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="goboard2.html#T18">GOBOARD2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_8"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E71:32_3_1_1_1_3_1_3_1_1_2"/><i><font color="Green">E514</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9"/>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1"/><i><font color="Green">E515</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1"/><i><font color="Green">E516</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1"/><i><font color="Green">E517</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1"/><b>then </b><i><font color="Green">E518</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_9_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green">E519</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E520</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green">E521</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green">E522</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E523</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E524</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green">E525</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>
 ;<br/>
<a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><i><font color="Green">E526</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)</span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E527</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E528</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/><b>then </b><i><font color="Green">E529</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/><i><font color="Green">E530</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1"/><i><font color="Green">E531</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/><i><font color="Green">E532</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/><b>then </b><i><font color="Green">E533</font></i>: 
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><i><font color="Green">E534</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b><i><font color="Green">E535</font></i>: 
( <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD2</font> &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b><i><font color="Green">E536</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b><i><font color="Green">E537</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD2</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a>, , , , , , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_9_1_1_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/><b>then </b><i><font color="Green">E538</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">MD2</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E20:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD2</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<b>hence </b><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_9_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D2</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:32_3_1_1_1_3_1_3_1_1_2_9_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_9_1"/>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_9"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E72:32_3_1_1_1_3_1_3_1_1_2"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10"><b>proof </b></a><div class="add">

<i><font color="Green">E539</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font>
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10"/><i><font color="Green">E540</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10"/><i><font color="Green">E541</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><i><font color="Green">E542</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , <a class="ref" href="jordan3.html#T27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_10"/>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E543</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> )
 ;<br/>

<a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green">E544</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">MD1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green">E545</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green">E546</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E547</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>
 ;<br/>
<a NAME="E9:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E548</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E10:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E549</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green">E550</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E551</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>
;<br/>
<a NAME="E14:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E15:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b><i><font color="Green">E552</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E16:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E553</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>
<a NAME="E17:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E554</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E555</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E19:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><i><font color="Green">E556</font></i>: 
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/><i><font color="Green">E557</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_1"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><i><font color="Green">E558</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E559</font></i>: 
( <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font> &amp; <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E3:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E560</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/>
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E5:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E561</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">MD1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E7:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1_1"><i><font color="Green">E63</font></i></a>, , , , , , , <a class="ref" href="jordan3.html#T31">JORDAN3:31</a></i>;<br/><b>hence </b><a NAME="E8:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2_1_1_2"/>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3_1_2"/>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">MD1</font>,<font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="integra1.html#T5">INTEGRA1:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">MD1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/><b>then </b>
 <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T10">INTEGRA1:10</a></i>;<br/>
<b>hence </b><a NAME="E22:32_3_1_1_1_3_1_3_1_1_2_10_3_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3_1_1_2_10_3"/>
 <a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font> <a href="hidden.html#R1">=</a>  <a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32_3_1_1_1_3_1_3_1_1_2_10"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15">lower_volume</a> <font color="Maroon">g</font>,<font color="Maroon">MD1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E73:32_3_1_1_1_3_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_3_1_1_1_3_1_3"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E34:32_3_1_1_1_3_1"/><i><font color="Green">E562</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_4"/><i><font color="Green">E563</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_4"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_4"/><b>then </b><i><font color="Green">E564</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_4"/><i><font color="Green">E565</font></i>: 
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)
 
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:32_3_1_1_1_3_1_4"/>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_3_1_1_1_3_1_4"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),1,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),1,<span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="integra2.html#T4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>) <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1_3_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:32_3_1_1_1_3_1"/><i><font color="Green">E566</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_1_3_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:32_3_1_1_1_3_1_5"/><i><font color="Green">E567</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E2:32_3_1_1_1_3_1_5"/><b>then </b><i><font color="Green">E568</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:32_3_1_1_1_3_1_5"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_3_1_1_1_3_1_5"/><b>then </b>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>)</span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8">INTEGRA1:def 8</a></i>;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="finseq_1.html#K17">|</a> <font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),1,<font color="Maroon">n1</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<b>then </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),1,<font color="Maroon">n1</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),1,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, , <a class="ref" href="integra2.html#T4">INTEGRA2:4</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>) <a href="finseq_1.html#K17">|</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:32_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan3.html#T25">JORDAN3:25</a></i>
;<br/>
<b>hence </b><a NAME="E7:32_3_1_1_1_3_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p3">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="integra1.html#D22">INTEGRA1:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E36:32_3_1_1_1_3_1"/><i><font color="Green">E569</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>)</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E37:32_3_1_1_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">n1</font>)</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D2</font>),<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">n1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="jordan3.html#K1">mid</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">D1</font>),<span class="p4">(<span class="default"><font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>)
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E38:32_3_1_1_1_3_1"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Olive">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:32_3_1_1_1"/><i><font color="Green">E570</font></i>: 
for <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="binarith.html#S1">BINARITH:sch 1</a>(<a class="ref" href="integra3.html#T2" target="_self">Th2</a>, );<br/></i>
<a NAME="E6:32_3_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="binarith.html#T5">BINARITH:5</a></i>;<br/>
<a NAME="E8:32_3_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:32_3_1_1_1"/>
 ex <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Olive">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Olive">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:32_3_1_1"/><i><font color="Green">E571</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="real_1.html#K5">-</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D1</font>,<font color="Maroon">j</font>) <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E5:32_3_1_1"/><i><font color="Green">E572</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E6:32_3_1_1"/><i><font color="Green">E573</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_3_1_1_2"/><i><font color="Green">E574</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 ;<br/>

<a NAME="E2:32_3_1_1_2"/><i><font color="Green">E575</font></i>: 
<font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra2.html#T1">INTEGRA2:1</a></i>;<br/>
<a NAME="E3:32_3_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:32_3_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:32_3_1_1_2"/><b>then </b>
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E6:32_3_1_1_2"/><b>then </b><i><font color="Green">E576</font></i>: 
<font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E7:32_3_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_3_1_1_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_3_1_1_2_1_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_2_1_1_1"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_3_1_1_2_1_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_3_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D1</font>,<font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_3_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:32_3_1_1"/><i><font color="Green">E577</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E8:32_3_1_1"/><i><font color="Green">E578</font></i>: 
(  <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="integra1.html#D21">INTEGRA1:def 21</a></i>;<br/>
<a NAME="E9:32_3_1_1"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E10:32_3_1_1"/><b>then </b>
<font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font> <a href="goboard1.html#K1">.</a> <span class="p1">(<span class="default"><a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E11:32_3_1_1"/><b>then </b>
 <a href="integra1.html#K17">indx</a> <font color="Maroon">D2</font>,<font color="Maroon">D1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>
 
<b>by </b><i>, , <a class="ref" href="goboard2.html#T19">GOBOARD2:19</a></i>;<br/>
<a NAME="E12:32_3_1_1"/><b>then </b>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">D2</font>, <a href="finseq_1.html#K3">len</a> <font color="Maroon">D2</font>) <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T2" target="_self">Th2</a>, <a class="ref" href="integra1.html#T45">INTEGRA1:45</a></i>;<br/>
<b>hence </b><a NAME="E13:32_3_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#T45">INTEGRA1:45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_3_1"/>
 ex <font color="Olive">D2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_3"/>
 ex <font color="Olive">D2</font> being   <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Olive">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Olive">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:32"/><i><font color="Green">E579</font></i>: 
for <font color="Olive">e</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:32_4"/><i><font color="Green">E580</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:32_4"/><b>then </b><i><font color="Green">E581</font></i>: 
<font color="Maroon">e</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E4:32_4"/><i><font color="Green">E582</font></i>: 
 <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>;<br/>
<a NAME="E5:32_4"/><i><font color="Green">E583</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="integra2.html#T36">INTEGRA2:36</a></i>;<br/>
<a NAME="E6:32_4"/>
not  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E7:32_4"/><b>then </b>
not  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:32_4"/><i><font color="Green">E584</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">D</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E11:32_4"/><i><font color="Green">E585</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">D</font> &amp; <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="integra3.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E12:32_4"/><i><font color="Green">E586</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">h</font> =  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  =  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">v</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E14:32_4"/><i><font color="Green">E591</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> holds <br/><font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1">FINSEQ_2:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">v1</font> being  <a href="integra2.html#V1">non-decreasing</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E16:32_4"/><i><font color="Green">E593</font></i>: 
<font color="Maroon">v</font>,<font color="Maroon">v1</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="integra2.html#T3">INTEGRA2:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 );<br/>
<a NAME="E17:32_4"/><i><font color="Green">E594</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">H</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:32_4_1"/><i><font color="Green">E595</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">H</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">H</font> )
 <b>by </b><i>, <a class="ref" href="rfinseq.html#T3">RFINSEQ:3</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:32_4_1"/><i><font color="Green">E596</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> &amp;  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="integra3.html#E4"><i><font color="Green">Lemma35</font></i></a>, </i>;<br/>
<a NAME="E5:32_4_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_4_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:32_4_1"/><b>then </b>
( <font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">v1</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E8:32_4_1"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:32_4"/><i><font color="Green">E597</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<a NAME="E20:32_4"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E21:32_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E22:32_4"/><b>then </b><i><font color="Green">E598</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E23:32_4"/>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E24:32_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E25:32_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E26:32_4"/><b>then </b><i><font color="Green">E599</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/>
<a NAME="E27:32_4"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_4_2_1"/>
(  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> or  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_2_1_1"><b>suppose </b></a><a NAME="E1:32_4_2_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_4_2_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_2_1_2"><b>suppose </b></a><a NAME="E1:32_4_2_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_4_2_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="integra3.html#E5:32"><i><font color="Green">E42</font></i></a>, <a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E29:32_4"/><i><font color="Green">E600</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E30:32_4"/><i><font color="Green">E601</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_4_3"/><i><font color="Green">E602</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:32_4_3"/><b>then </b><i><font color="Green">E603</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">D1</font> = <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> as    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font> ;<br/>
<b>consider </b><font color="Maroon">D2</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:32_4_3"/><i><font color="Green">E604</font></i>: 
( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:32_4_3"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E2"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="rfunct_1.html#D11">RFUNCT_1:def 11</a></i>;<br/>
<a NAME="E7:32_4_3"/><b>then </b>
 <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, <a class="ref" href="integra1.html#T48">INTEGRA1:48</a></i>;<br/>
<a NAME="E8:32_4_3"/><b>then </b><i><font color="Green">E605</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E9:32_4_3"/><i><font color="Green">E606</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E10:32_4_3"/><i><font color="Green">E607</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="integra2.html#D3">INTEGRA2:def 3</a></i>;<br/>
<a NAME="E11:32_4_3"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E12:32_4_3"/><b>then </b><i><font color="Green">E608</font></i>: 
 <a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:32_4_3"/><i><font color="Green">E609</font></i>: 
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_4_3_1"/><i><font color="Green">E610</font></i>: 
for <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/> <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Olive">n1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:32_4_3_1_1"/><i><font color="Green">E611</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:32_4_3_1_1"/><b>then </b><i><font color="Green">E612</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_4_3_1_1_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_4_3_1_1_1_1"/><i><font color="Green">E613</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:32_4_3_1_1_1_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_4_3_1_1_1_1"/><b>then </b>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><b>hence </b><a NAME="E4:32_4_3_1_1_1_1"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_4_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_4_3_1_1_1_2"/><i><font color="Green">E614</font></i>: 
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:32_4_3_1_1_1_2"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="integra1.html#D5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:32_4_3_1_1_1_2"/><b>then </b><i><font color="Green">E615</font></i>: 
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#D6">INTEGRA1:def 6</a></i>;<br/><a NAME="E4:32_4_3_1_1_1_2"/><i><font color="Green">E616</font></i>: 
<font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="integra1.html#T9">INTEGRA1:9</a></i>;<br/><a NAME="E5:32_4_3_1_1_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E6:32_4_3_1_1_1_2"/><b>then </b>
<font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E7:32_4_3_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">D</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, , <a class="ref" href="goboard1.html#D1">GOBOARD1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:32_4_3_1_1_1_2"/>
 <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:32_4_3_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:32_4_3_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:32_4_3_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:32_4_3_1"/><b>then </b><i><font color="Green">E617</font></i>: 
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:32_4_3_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:32_4_3_1"/><i><font color="Green">E618</font></i>: 
( <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">n1</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E9:32_4_3_1"/><i><font color="Green">E619</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:32_4_3_1"/><b>then </b><i><font color="Green">E620</font></i>: 
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">n1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:32_4_3_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:32_4_3_1"/>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="integra3.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:32_4_3"/>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_4_3_2"/><i><font color="Green">E621</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_4_3_2_1"/><i><font color="Green">E622</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:32_4_3_2_1"/><i><font color="Green">E623</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<a NAME="E3:32_4_3_2_1"/><i><font color="Green">E624</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:32_4_3_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:32_4_3_2_1"/><b>then </b><i><font color="Green">E625</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:32_4_3_2_1"/><b>then </b><i><font color="Green">E626</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:32_4_3_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:32_4_3_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:32_4_3_2_1"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E10:32_4_3_2_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_4_3_2"/>
(  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> holds <br/> <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:32_4_3_2"/><i><font color="Green">E627</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> &amp;  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E5:32_4_3_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:32_4_3_2"/><b>then </b><i><font color="Green">E628</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E7:32_4_3_2"/><b>then </b>
 <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<a NAME="E8:32_4_3_2"/><b>then </b><i><font color="Green">E629</font></i>: 
<font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:32_4_3_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:32_4_3_2"/><b>then </b><i><font color="Green">E630</font></i>: 
 <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:32_4_3_2"/><i><font color="Green">E631</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:32_4_3_2"/><i><font color="Green">E633</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> &amp;  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">m1</font> )
 <b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E14:32_4_3_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E15:32_4_3_2"/><b>then </b><i><font color="Green">E634</font></i>: 
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="integra2.html#T2">INTEGRA2:2</a></i>;<br/>
<a NAME="E16:32_4_3_2"/>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:32_4_3_2"/><i><font color="Green">E635</font></i>: 
( <font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v</font> &amp; <font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k2</font> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E19:32_4_3_2"/><i><font color="Green">E636</font></i>: 
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E20:32_4_3_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2">divset</a> <font color="Maroon">D</font>,<font color="Maroon">k2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E21:32_4_3_2"/><b>then </b><i><font color="Green">E637</font></i>: 
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k2</font>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T22">INTEGRA1:22</a></i>;<br/>
<a NAME="E22:32_4_3_2"/>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D7">INTEGRA1:def 7</a></i>;<br/>
<a NAME="E23:32_4_3_2"/><b>then </b>
<font color="Maroon">k2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E24:32_4_3_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E25:32_4_3_2"/><b>then </b>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_circ.html#D4">PRE_CIRC:def 4</a></i>;<br/>
<b>hence </b><a NAME="E26:32_4_3_2"/>
<font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K3">min</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K14">upper_volume</a> <span class="p3">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">D3</font> being    <a href="integra1.html#M3">Element</a> of  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E16:32_4_3"/><i><font color="Green">E638</font></i>: 
( <font color="Maroon">D</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D3</font> &amp; <font color="Maroon">D1</font> <a href="integra1.html#R4">&lt;=</a> <font color="Maroon">D3</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D3</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="integra3.html#E3:32"><i><font color="Green">E40</font></i></a>, </i>;<br/>
<a NAME="E17:32_4_3"/><i><font color="Green">E639</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K16">delta</a> <font color="Maroon">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:32"><i><font color="Green">E59</font></i></a>, , </i>;<br/>
<a NAME="E18:32_4_3"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E19:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E20:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E7:32"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E21:32_4_3"/><b>then </b><i><font color="Green">E640</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra3.html#T3" target="_self">Th3</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E22:32_4_3"/><i><font color="Green">E641</font></i>: 
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E23:32_4_3"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E24:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E25:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:32"><i><font color="Green">E54</font></i></a>, <a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E26:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p0">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
;<br/>
<a NAME="E27:32_4_3"/><b>then </b><i><font color="Green">E642</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_2.html#T178">REAL_2:178</a></i>;<br/>
<b>set </b><font color="Maroon">s</font> =  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">sD</font> =  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font>;<br/>
<b>set </b><font color="Maroon">sD1</font> =  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">sD2</font> =  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font>;<br/>
<a NAME="E28:32_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="integra3.html#T2" target="_self">Th2</a>, , <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E29:32_4_3"/><b>then </b><i><font color="Green">E647</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E30:32_4_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span>
 
;<br/>
<a NAME="E31:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E32:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K2">sup</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <span class="p5">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K2">delta</a> <font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E33:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="integra3.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">D</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E35:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E36:32_4_3"/><b>then </b><i><font color="Green">E648</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>;<br/>
<a NAME="E37:32_4_3"/>
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="integra1.html#K8">divs</a> <font color="Maroon">A</font>
 
;<br/>
<a NAME="E38:32_4_3"/><b>then </b><i><font color="Green">E649</font></i>: 
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E39:32_4_3"/>
<span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K7">lower_sum</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra2.html#D5">INTEGRA2:def 5</a></i>;<br/>
<a NAME="E40:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#D12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E41:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E42:32_4_3"/><b>then </b>
 <a href="pscomp_1.html#K2">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="integra1.html#K10">lower_sum_set</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E43:32_4_3"/><b>then </b>
 <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D16">INTEGRA1:def 16</a></i>;<br/>
<a NAME="E44:32_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E45:32_4_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E46:32_4_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<b>hence </b><a NAME="E47:32_4_3"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 ;<br/>


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

<b>thus </b><a NAME="E31:32_4"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


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

<b>hence </b><a NAME="E11:32"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><a href="integra2.html#K4">lower_sum</a> <font color="Maroon">f</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="integra1.html#K12">lower_integral</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="seq_2.html#D7">SEQ_2:def 7</a></i>;<br/>


</div>
