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

<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font> holds   <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Olive" title="b1">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be    <a href="integra1.html#M1" title="INTEGRA1:mode.1">Division</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span><br/>



<a NAME="E1:40"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span><br/>

<a NAME="E3:40_1"/>
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>;<br/>
<a NAME="E4:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> (<font color="Maroon" title="c2">D</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T19" target="_self" title="INTEGRA1:th.19">Th21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="integra1.html#T20" target="_self" title="INTEGRA1:th.20">Th22</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:40_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D7" target="_self" title="INTEGRA1:def.7">Def8</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#D6" target="_self" title="INTEGRA1:def.6">Def7</a></span>
;<br/>
<a NAME="E3:40"/><span class="kw">then </span>
 <a href="integra1.html#K5" title="INTEGRA1:func.5">lower_volume</a> (<span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p1">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K4" title="INTEGRA1:func.4">upper_volume</a> (<span class="p2">(<span class="default"><a href="rfunct_1.html#K7" title="RFUNCT_1:func.7">chi</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span>,<font color="Maroon" title="c2">D</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="integra1.html#T23" target="_self" title="INTEGRA1:th.23">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
