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<b>let </b><font color="Maroon" title="c1">y</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">A</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E19">A1</font></i>: 
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon" title="c1">y</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> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">D3</font> being    <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="c2">A</font><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c4">D3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">D3</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E7:18"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D3</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/>
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E8:18"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#T8" title="INTEGRA1:th.8">INTEGRA1:8</a></i>;<br/><a NAME="E3:18_1_1_1"/><b>then </b>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="integra2.html#T1" title="INTEGRA2:th.1">INTEGRA2:1</a></i>;<br/><a NAME="E4:18_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_1"/>
 ex <font color="Olive" title="b1">D</font> being   <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="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being   <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="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_2_1"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1: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">D3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E2:18_1_1_2_1"/><i><font color="Green" title="E25">A9</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</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">D3</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18"><i><font color="Green" title="E21">A3</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="E3:18_1_1_2_1"/>
 <a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="integra1.html#T11" title="INTEGRA1:th.11">INTEGRA1:11</a></i>;<br/>
<a NAME="E4:18_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>;<br/>
<a NAME="E5:18_1_1_2_1"/><b>then </b><i><font color="Green" title="E26">A10</font></i>: 
 <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/>
<a NAME="E6:18_1_1_2_1"/><b>then </b><i><font color="Green" title="E27">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">D</font> = <font color="Maroon" title="c4">D3</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 as  <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="ref" href="integra1.html#T36" title="INTEGRA1:th.36">INTEGRA1:36</a></i>;<br/>
<a NAME="E8:18_1_1_2_1"/><i><font color="Green" title="E28">A12</font></i>: 
(  <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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; ( for <font color="Olive" title="b1">m</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">m</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E9:18_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="integra3.html#E5:18_1_1_2_1"><i><font color="Green" title="E26">A10</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="c6">D</font> = <font color="Maroon" title="c5">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="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<a NAME="E11:18_1_1_2_1"/>
(  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D3</font> &amp;  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c4">D3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T36" title="FINSEQ_5:th.36">FINSEQ_5:36</a>, <a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E12:18_1_1_2_1"/><b>then </b><i><font color="Green" title="E29">A13</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c6">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:18_1_1_2_1"/><i><font color="Green" title="E30">A14</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/>
<a NAME="E14:18_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">D</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="c6">D</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/>
<a NAME="E15:18_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">D</font> <a href="seq_1.html#K1" title="SEQ_1: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="c6">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1"><i><font color="Green" title="E24">A8</font></i></a>, <a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="integra3.html#E13:18_1_1_2_1"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E16:18_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c6">D</font> is    <a href="integra1.html#M1" title="INTEGRA1:mode.1">DivisionPoint</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E12:18_1_1_2_1"><i><font color="Green" title="E29">A13</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="c7">D</font> = <font color="Maroon" title="c6">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="c2">A</font> <b>by </b><i><a class="ref" href="integra1.html#D4" title="INTEGRA1:def.4">INTEGRA1:def 4</a></i>;<br/>
<a NAME="E18:18_1_1_2_1"/>
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="integra1.html#K8" title="INTEGRA1:func.8">divs</a> <font color="Maroon" title="c2">A</font>
 
;<br/>
<a NAME="E19:18_1_1_2_1"/><b>then </b><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>
<a NAME="E20:18_1_1_2_1"/><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E33">A17</font></i>: <font color="Maroon" title="c1">y</font> = 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font>
<b>by </b><i><a class="txt" href="integra3.html#E4:18"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfinseq.html#T21" title="RFINSEQ:th.21">RFINSEQ:21</a></i>
;<br/>
<a NAME="E2:18_1_1_2_1_1"/><i><font color="Green" title="E34">A18</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_2_1_1_2"/><i><font color="Green" title="E35">A19</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E2:18_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E36">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E3:18_1_1_2_1_1_2"/><i><font color="Green" title="E37">A21</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_2"><i><font color="Green" title="E35">A19</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_2"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E3:18_1_1_2_1_1_2"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_2"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E35">A22</font></i>:  <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="c4">D3</font>,1</span>)</span> = 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="integra1.html#D6" title="INTEGRA1:def.6">INTEGRA1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E8:18"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E8:18"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green" title="E23">A7</font></i></a></i>
;<br/>
<a NAME="E4:18_1_1_2_1_1"/><i><font color="Green" title="E36">A23</font></i>: 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <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> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a>, <a class="txt" href="integra3.html#E8:18"><i><font color="Green" title="E22">A5</font></i></a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_4"><b>proof </b></a><div class="add">

<i><font color="Green" title="E37">A24</font></i>: 2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
2 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
1

;<br/>
<a NAME="E2:18_1_1_2_1_1_4"/><i><font color="Green" title="E38">A25</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:18_1_1_2_1_1_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_4"/><b>then </b><i><font color="Green" title="E39">A26</font></i>: 
 <a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4"><i><font color="Green" title="E37">A24</font></i></a>, <a class="ref" href="jordan3.html#T26" title="JORDAN3:th.26">JORDAN3:26</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_4"/><i><font color="Green" title="E40">A27</font></i>: 
( 2 <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#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E6:18_1_1_2_1_1_4"/><i><font color="Green" title="E41">A28</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_4_2"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4"><i><font color="Green" title="E40">A27</font></i></a>, <a class="ref" href="jordan3.html#T27" title="JORDAN3:th.27">JORDAN3:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 2</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
<b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4"><i><font color="Green" title="E38">A25</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1

;<br/>
<b>hence </b><a NAME="E2:18_1_1_2_1_1_4_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18_1_1_2_1_1_4"/>
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="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></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#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">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="18_1_1_2_1_1_4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:18_1_1_2_1_1_4_3"/><i><font color="Green" title="E42">A29</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:18_1_1_2_1_1_4_3"/><i><font color="Green" title="E43">A30</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">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="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E3:18_1_1_2_1_1_4_3"/><b>then </b><i><font color="Green" title="E44">A31</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">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="c7">D</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1_1_4"><i><font color="Green" title="E41">A28</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_4_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 )
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_4_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 
;<br/>
<b>then </b><i><font color="Green" title="E45">A32</font></i>: <span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4"><i><font color="Green" title="E40">A27</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="jordan3.html#T31" title="JORDAN3:th.31">JORDAN3:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/>
<a NAME="E7:18_1_1_2_1_1_4_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font> )
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:18_1_1_2_1_1_4_3"/><b>then </b><i><font color="Green" title="E46">A33</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:18_1_1_2_1_1_4_3"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D3</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:18_1_1_2_1_1_4_3"/><b>then </b><i><font color="Green" title="E47">A34</font></i>: 
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">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> <span class="p2">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="integra1.html#K3" title="INTEGRA1:func.3">vol</a> <span class="p2">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1_1_4_3"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>
<a NAME="E11:18_1_1_2_1_1_4_3"/><i><font color="Green" title="E48">A35</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:18_1_1_2_1_1_4_3_2"/><i><font color="Green" title="E49">A36</font></i>: 
1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E2:18_1_1_2_1_1_4_3_2"/>
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">D3</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18_1_1_2_1_1_4_3"><i><font color="Green" title="E46">A33</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="E3:18_1_1_2_1_1_4_3_2"/><b>then </b><i><font color="Green" title="E50">A37</font></i>: 
(  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/>
<a NAME="E4:18_1_1_2_1_1_4_3_2"/><i><font color="Green" title="E51">A38</font></i>: 
<font color="Maroon" title="c8">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="c7">D</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:18_1_1_2_1_1_4_3_2"/><b>then </b><i><font color="Green" title="E52">A39</font></i>: 
<font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_2_1_1_4_3_2_1"/>
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_4_3_2_1_1"/><i><font color="Green" title="E53">A40</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_4_3_2_1_1"/><b>then </b><i><font color="Green" title="E54">A41</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:18_1_1_2_1_1_4_3_2_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3_2_1_1"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_2_1_1_4_3_2_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3_2_1_1"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1_1_4_3_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_1_1_4_3_2_1_2"/><i><font color="Green" title="E53">A42</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_1_2_1_1_4_3_2_1_2"/><b>then </b><i><font color="Green" title="E54">A43</font></i>: 
(  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:18_1_1_2_1_1_4_3_2_1_2"/>
( <font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="txt" href="integra3.html#E1:18_1_1_2_1_1_4_3_2_1_2"><i><font color="Green" title="E53">A42</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E4:18_1_1_2_1_1_4_3_2_1_2"/><b>then </b>
( <font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font> &amp; <font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>then </b><font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> = 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>

;<br/><a NAME="E6:18_1_1_2_1_1_4_3_2_1_2"/><b>then </b>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p3">(<span class="default"><a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E50">A37</font></i></a>, <a class="txt" href="integra3.html#E5:18_1_1_2_1_1_4_3_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1_4_3_2_1_2"><i><font color="Green" title="E54">A43</font></i></a>, <a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><b>hence </b><a NAME="E7:18_1_1_2_1_1_4_3_2_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c4">D3</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c7">D</font>,<font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:18_1_1_2_1_1_4_3"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra3.html#E3:18_1_1_2_1_1_4_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:18_1_1_2_1_1_4_3"/><b>then </b>
<font color="Maroon" title="c8">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="c7">D</font>
 
<b>by </b><i><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="E14:18_1_1_2_1_1_4_3"/>
<span class="p1">(<span class="default"><a href="jordan3.html#K1" title="JORDAN3:func.1">mid</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span>,2,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra3.html#E10:18_1_1_2_1_1_4_3"><i><font color="Green" title="E47">A34</font></i></a>, <a class="txt" href="integra3.html#E11:18_1_1_2_1_1_4_3"><i><font color="Green" title="E48">A35</font></i></a>, <a class="ref" href="integra1.html#D8" title="INTEGRA1:def.8">INTEGRA1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_1_1_2_1_1_4"/>
<span class="p1">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">D3</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E4:18_1_1_2_1_1_4"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="integra3.html#E6:18_1_1_2_1_1_4"><i><font color="Green" title="E41">A28</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><font color="Maroon" title="c1">y</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="integra1.html#K15" title="INTEGRA1:func.15">lower_volume</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2_1_1"><i><font color="Green" title="E33">A17</font></i></a>, <a class="txt" href="integra3.html#E2:18_1_1_2_1_1"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="integra3.html#E3:18_1_1_2_1_1"><i><font color="Green" title="E35">A22</font></i></a>, <a class="txt" href="integra3.html#E4:18_1_1_2_1_1"><i><font color="Green" title="E36">A23</font></i></a>, <a class="ref" href="rvsum_1.html#T106" title="RVSUM_1:th.106">RVSUM_1:106</a></i>
<br/>.= 
 <a href="integra1.html#K7" title="INTEGRA1:func.7">lower_sum</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c7">D</font>
<b>by </b><i><a class="ref" href="integra1.html#D10" title="INTEGRA1:def.10">INTEGRA1:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E7:18_1_1_2_1_1"/>
<font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">D</font>
 <b>by </b><i><a class="txt" href="integra3.html#E19:18_1_1_2_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="integra1.html#D12" title="INTEGRA1:def.12">INTEGRA1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:18_1_1_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E22:18_1_1_2_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T34" title="FINSEQ_3:th.34">FINSEQ_3:34</a></i>;<br/>
<b>then </b><i><font color="Green" title="E33">A44</font></i>: <font color="Maroon" title="c7">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 = 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="rfinseq.html#D2" title="RFINSEQ:def.2">RFINSEQ:def 2</a></i>
<br/>.= 
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2

;<br/>
<a NAME="E24:18_1_1_2_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">D3</font>
 
<b>by </b><i><a class="txt" href="integra3.html#E6:18_1_1_2_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E25:18_1_1_2_1"/><b>then </b>
2 <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">D3</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E26:18_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">D3</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2
 
<b>by </b><i><a class="txt" href="integra3.html#E8:18"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E27:18_1_1_2_1"/>
 ex <font color="Olive" title="b1">D</font> being   <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="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="integra3.html#E1:18_1_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="integra3.html#E19:18_1_1_2_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="integra3.html#E20:18_1_1_2_1"><i><font color="Green" title="E32">A16</font></i></a>, <a class="txt" href="integra3.html#E23:18_1_1_2_1"><i><font color="Green" title="E33">A44</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:18_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being   <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="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:18"/>
 ex <font color="Olive" title="b1">D</font> being   <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="c2">A</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="integra1.html#K10" title="INTEGRA1:func.10">lower_sum_set</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b1">D</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>


</div>
