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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="integra1.html#V1" target="_self">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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="integra1.html#M2" target="_self">Division</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="integra1.html#M3" target="_self">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>







<b>assume </b><a NAME="E1:86"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="rfunct_1.html#R2">is_bounded_below_on</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:86"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E3:86"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:86"/><i><font color="Green">E70</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E5:86"/><i><font color="Green">E71</font></i>: 
 <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E2:86"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E6:86"/><i><font color="Green">E72</font></i>: 
 <a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E3:86"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E7:86"/><i><font color="Green">E73</font></i>: 
 <a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E4:86"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E8:86"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="integra1.html#E5:86"><i><font color="Green">E71</font></i></a>, <a class="txt" href="integra1.html#E6:86"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_2.html#T140">FINSEQ_2:140</a></i>;<br/>
<a NAME="E9:86"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:86_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:86_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="integra1.html#E1:86"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E1:86_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="integra1.html#T56" target="_self">Th56</a></i>;<br/>
<b>hence </b><a NAME="E3:86_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="integra1.html#E5:86"><i><font color="Green">E71</font></i></a>, <a class="txt" href="integra1.html#E6:86"><i><font color="Green">E72</font></i></a>, <a class="ref" href="rvsum_1.html#T27">RVSUM_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:86"/><b>then </b>
 <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" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:86"><i><font color="Green">E73</font></i></a>, <a class="txt" href="integra1.html#E8:86"><i><font color="Green">E74</font></i></a>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<a NAME="E11:86"/><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="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" target="_self">lower_volume</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:86"><i><font color="Green">E71</font></i></a>, <a class="txt" href="integra1.html#E6:86"><i><font color="Green">E72</font></i></a>, <a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>;<br/>
<a NAME="E12:86"/><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="integra1.html#K15" target="_self">lower_volume</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</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" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E13:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K9">+</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" target="_self">lower_volume</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<b>hence </b><a NAME="E14:86"/>
<span class="p1">(<span class="default"><a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K7" target="_self">lower_sum</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="integra1.html#K7" target="_self">lower_sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K6">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div>
