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

<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="integra1.html#V1" title="INTEGRA1:attr.1">closed-interval</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c3">D</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:16"/><i><font color="Green" title="E16">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E4:16"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> )
 
<b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:16_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="integra6.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">j</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:16_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:16_1"/><b>then </b>
<font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="integra6.html#E2:16_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:16_1"/>
<font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="integra6.html#E2:16_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">D</font> = <font color="Maroon" title="c3">D</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="goboard1.html#V1" title="GOBOARD1:attr.1">increasing</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="integra6.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E6:16"/>
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D</font> &amp; <font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:16_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="integra6.html#E3:16_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:16_2"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="integra6.html#E3:16_2"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/>
<a NAME="E6:16_2"/><i><font color="Green" title="E19">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">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="c4">D</font> &amp; <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="integra6.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="integra6.html#E3:16_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E7:16_2"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/>
<a NAME="E8:16_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="integra6.html#E6:16_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E9:16_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E10:16_2"/><b>then </b>
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra6.html#E7:16_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/>
<b>hence </b><a NAME="E11:16_2"/>
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="integra6.html#E5:16_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:16"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:16"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E10:16"/><b>then </b>
<font color="Maroon" title="c4">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra6.html#E1:16"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
<font color="Maroon" title="c4">D</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">DivisionPoint</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="integra6.html#E7:16"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c4">D</font> as    <a href="integra1.html#M3" title="INTEGRA1:mode.3">Element</a> of  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="integra1.html#D3" title="INTEGRA1:def.3">INTEGRA1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">D</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:16_3"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font>
 ;<br/><a NAME="E2:16_3"/><b>then </b>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:16_3"/>
<font color="Maroon" title="c5">D</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:16"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">D</font> holds <br/><font color="Maroon" title="c5">D</font> <a href="goboard1.html#K1" title="GOBOARD1: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="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="integra6.html#E3:16"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>


</div>
