<?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">D1</font>, <font color="Maroon" title="c3">D2</font> be    <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>;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="integra1.html#R4" title="INTEGRA1:pred.4">&lt;=</a> <font color="Maroon" title="c3">D2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">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="E2:43"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">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">D2</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <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="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> );<br/>
<b>consider </b><font color="Maroon" title="c5">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">D2</font></span>)</span><b> such that </b><br/><a NAME="E4:43"/><i><font color="Green" title="E29">A3</font></i>: 
for <font color="Olive" title="b1">x1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> iff ( <font color="Olive" title="b1">x1</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">D2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1" title="SUBSET_1:sch.1">SUBSET_1:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <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="E6:43"/>
not <font color="Maroon" title="c6">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1"/>
 ex <font color="Olive" title="b1">n</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 <br/>( <font color="Olive" title="b1">n</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">D2</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</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="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_1"/><i><font color="Green" title="E30">A4</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</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">D2</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span> <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="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</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="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E2:43_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</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="c3">D2</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/>
<b>hence </b><a NAME="E3:43_1_1_1"/><i><font color="Green" title="E31">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</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">D2</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="E4:43_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</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="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E5:43_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</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="c2">D1</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="E6:43_1_1_1"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">D1</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="E7:43_1_1_1"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</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="c1">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="E8:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">D1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1_1"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span> <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="c2">D1</font>
 <b>by </b><i><a class="txt" href="integra2.html#E6:43_1_1_1"><i><font color="Green" title="E32">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>

<a NAME="E10:43_1_1_1"/>
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">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="E11:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">j</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="c3">D2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E12:43_1_1_1"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="integra2.html#E3:43_1_1_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font>
;<br/>

<b>thus </b><a NAME="E2:43_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</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">D2</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span> <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="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E1:43_1_1"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_1"/>
not <font color="Maroon" title="c6">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="integra2.html#E4:43"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E8:43"/><i><font color="Green" title="E30">A7</font></i>: 
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/>
<a NAME="E9:43"/><b>then </b><i><font color="Green" title="E31">A8</font></i>: 
(  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</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">D2</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</font></span>)</span> <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="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E4:43"><i><font color="Green" title="E29">A3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">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="E11:43"/><i><font color="Green" title="E32">A9</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="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E12:43"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E4:43"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="integra2.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a></i>;<br/>
<a NAME="E13:43"/><i><font color="Green" title="E34">A11</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_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="43_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_2_1_1_1"/>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:43_2_1_1_1"/><b>then </b>
(  <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="c2">D1</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="c1">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="c2">D1</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:43_2_1_1_1"/><b>then </b><i><font color="Green" title="E35">A12</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font> <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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_1_1_1_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c4">j</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="43_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:43_2_1_1_1_1_1_1"/><b>then </b>
(  <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="c3">D2</font>,<font color="Maroon" title="c4">j</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="c1">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="c3">D2</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:43_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E36">A13</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><a NAME="E4:43_2_1_1_1_1_1_1"/><i><font color="Green" title="E37">A14</font></i>: 
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_2_1_1_1_1_1_1_1"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</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="c2">D1</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:43_2_1_1_1_1_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D1</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="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E3:43_2_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</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="integra2.html#E1:43_2_1_1_1_1_1_1_1"><i><font color="Green" title="E38">A15</font></i></a></i>;<br/>
<a NAME="E4:43_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">A</font> <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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E5:43_2_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E3:43_2_1_1_1_1_1_1_1"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E6:43_2_1_1_1_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> &amp;  <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> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
;<br/>
<a NAME="E7:43_2_1_1_1_1_1_1_1"/><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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E8:43_2_1_1_1_1_1_1_1"/>
 <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:43_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:43_2_1_1_1_1_1_1_2"/><i><font color="Green" title="E38">A17</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</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="c3">D2</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:43_2_1_1_1_1_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D2</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="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E3:43_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E39">A18</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</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="integra2.html#E1:43_2_1_1_1_1_1_1_2"><i><font color="Green" title="E38">A17</font></i></a></i>;<br/>
<a NAME="E4:43_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c1">A</font> <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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E5:43_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E3:43_2_1_1_1_1_1_1_2"><i><font color="Green" title="E39">A18</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E6:43_2_1_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> &amp;  <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> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
;<br/>
<a NAME="E7:43_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E9:43"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:43_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:43_2_1_1_1_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra2.html#E3:43_2_1_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="integra2.html#E3:43_2_1_1_1_1_1_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="integra2.html#E4:43_2_1_1_1_1_1_1"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_2_1_1_1_1_1_2"/><i><font color="Green" title="E36">A19</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:43_2_1_1_1_1_1_2"/><b>then </b>
(  <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="c3">D2</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c3">D2</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:43_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A20</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <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"><font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><a NAME="E4:43_2_1_1_1_1_1_2"/><i><font color="Green" title="E38">A21</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_2_1_1_1_1_1_2_1"/><i><font color="Green" title="E39">A22</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c3">D2</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="integra2.html#E1:43_2_1_1_1_1_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>
<a NAME="E2:43_2_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E40">A23</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <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="c3">D2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:43_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c1">A</font> <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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E4:43_2_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E41">A24</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E5:43_2_1_1_1_1_1_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D2</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="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E6:43_2_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43_2_1_1_1_1_1_2_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="txt" href="integra2.html#E4:43_2_1_1_1_1_1_2_1"><i><font color="Green" title="E41">A24</font></i></a></i>;<br/>
<a NAME="E7:43_2_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E42">A25</font></i>: 
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp;  <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> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
;<br/>
<a NAME="E8:43_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E9:43_2_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E10:43_2_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="integra2.html#E1:43_2_1_1_1_1_1_2_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E11:43_2_1_1_1_1_1_2_1"/><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> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E9:43"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E7:43_2_1_1_1_1_1_2_1"><i><font color="Green" title="E42">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:43_2_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E13:43_2_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:43_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:43_2_1_1_1_1_1_2_2"/><i><font color="Green" title="E39">A26</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</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="c3">D2</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:43_2_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c1">A</font> <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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/>
<a NAME="E3:43_2_1_1_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E40">A27</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E4:43_2_1_1_1_1_1_2_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">D2</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="integra1.html#D2" title="INTEGRA1:def.2">INTEGRA1:def 2</a></i>;<br/>
<a NAME="E5:43_2_1_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E1:43_2_1_1_1_1_1_2_2"><i><font color="Green" title="E39">A26</font></i></a>, <a class="txt" href="integra2.html#E3:43_2_1_1_1_1_1_2_2"><i><font color="Green" title="E40">A27</font></i></a></i>;<br/>
<a NAME="E6:43_2_1_1_1_1_1_2_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> &amp;  <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> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <font color="Maroon" title="c1">A</font> )
 
;<br/>
<a NAME="E7:43_2_1_1_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="integra2.html#E9:43"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:43_2_1_1_1_1_1_2_2"/>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:43_2_1_1_1_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra2.html#E3:43_2_1_1_1"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="integra2.html#E3:43_2_1_1_1_1_1_2"><i><font color="Green" title="E37">A20</font></i></a>, <a class="txt" href="integra2.html#E4:43_2_1_1_1_1_1_2"><i><font color="Green" title="E38">A21</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:43_2_1_1_1"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_2_1_1_2"/><i><font color="Green" title="E35">A28</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="E2:43_2_1_1_2"/><b>then </b>
(  <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="c2">D1</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c2">D1</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E3:43_2_1_1_2"/><b>then </b><i><font color="Green" title="E36">A29</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font> <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"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><a NAME="E4:43_2_1_1_2"/><i><font color="Green" title="E37">A30</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2_1_1_2_1"/><i><font color="Green" title="E38">A31</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:43_2_1_1_2_1"/>
<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="c2">D1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</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="E3:43_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E39">A32</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="c2">D1</font> )
 
<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="E4:43_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="integra2.html#E1:43_2_1_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:43_2_1_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &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> <font color="Maroon" title="c2">D1</font> )
 
<b>by </b><i><a class="txt" href="integra2.html#E3:43_2_1_1_2_1"><i><font color="Green" title="E39">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:43_2_1_1_2_1"/><b>then </b>
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="c2">D1</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="E7:43_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E41">A34</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="c2">D1</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="E8:43_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E42">A35</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <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="c2">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:43_2_1_1_2_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D1</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="c3">D2</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:43"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">n</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="E11:43_2_1_1_2_1"/><i><font color="Green" title="E43">A36</font></i>: 
( <font color="Maroon" title="c9">n</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">D2</font> &amp; <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E8:43_2_1_1_2_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E12:43_2_1_1_2_1"/>
<font color="Maroon" title="c9">n</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="c3">D2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra2.html#E11:43_2_1_1_2_1"><i><font color="Green" title="E43">A36</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="E13:43_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E44">A37</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</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="c3">D2</font> )
 
<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="E14:43_2_1_1_2_1"/><b>then </b>
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="c3">D2</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:43_2_1_1_2_1"/><b>then </b>
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="c3">D2</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="E16:43_2_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="c3">D2</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="E17:43_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:43_2_1_1_2_1"><i><font color="Green" title="E38">A31</font></i></a>, <a class="txt" href="integra2.html#E11:43_2_1_1_2_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="txt" href="integra2.html#E13:43_2_1_1_2_1"><i><font color="Green" title="E44">A37</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E18:43_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E4:43"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="integra2.html#E8:43_2_1_1_2_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="txt" href="integra2.html#E11:43_2_1_1_2_1"><i><font color="Green" title="E43">A36</font></i></a></i>;<br/>
<a NAME="E19:43_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/>
<a NAME="E20:43_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E11:43_2_1_1_2_1"><i><font color="Green" title="E43">A36</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<b>hence </b><a NAME="E21:43_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E4:43_2_1_1_2_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="integra2.html#E7:43_2_1_1_2_1"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:43_2_1_1_2"/><b>then </b>
(  <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="c3">D2</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c3">D2</font>,<font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="integra1.html#D5" title="INTEGRA1:def.5">INTEGRA1:def 5</a></i>;<br/><a NAME="E6:43_2_1_1_2"/><b>then </b><i><font color="Green" title="E38">A38</font></i>: 
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <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"><font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="integra1.html#T5" title="INTEGRA1:th.5">INTEGRA1:5</a></i>;<br/><a NAME="E7:43_2_1_1_2"/><i><font color="Green" title="E39">A39</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c3">D2</font>
 <b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="integra2.html#E4:43_2_1_1_2"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/><a NAME="E8:43_2_1_1_2"/>
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:43_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><a NAME="E10:43_2_1_1_2"/><b>then </b><i><font color="Green" title="E40">A40</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="integra2.html#E7:43_2_1_1_2"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/><a NAME="E11:43_2_1_1_2"/><b>then </b><i><font color="Green" title="E41">A41</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra2.html#E12:43"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:43_2_1_1_2"/><i><font color="Green" title="E42">A42</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2_1_1_2_2"/><i><font color="Green" title="E43">A43</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:43_2_1_1_2_2"/><i><font color="Green" title="E44">A44</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">D1</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="c3">D2</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E1:43"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="integra1.html#D20" title="INTEGRA1:def.20">INTEGRA1:def 20</a></i>;<br/>
<a NAME="E3:43_2_1_1_2_2"/><i><font color="Green" title="E45">A45</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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="c2">D1</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E1:43_2_1_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="integra1.html#T9" title="INTEGRA1:th.9">INTEGRA1:9</a></i>;<br/>
<a NAME="E4:43_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E46">A46</font></i>: 
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <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="c2">D1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">n</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="E6:43_2_1_1_2_2"/><i><font color="Green" title="E47">A47</font></i>: 
( <font color="Maroon" title="c9">n</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">D2</font> &amp; <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">n</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E2:43_2_1_1_2_2"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E7:43_2_1_1_2_2"/>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<b>by </b><i><a class="txt" href="integra2.html#E7:43_2_1_1_2"><i><font color="Green" title="E39">A39</font></i></a>, <a class="txt" href="integra2.html#E1:43_2_1_1_2_2"><i><font color="Green" title="E43">A43</font></i></a>, <a class="txt" href="integra2.html#E6:43_2_1_1_2_2"><i><font color="Green" title="E47">A47</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E8:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E9:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E10:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="integra2.html#E6:43_2_1_1_2_2"><i><font color="Green" title="E47">A47</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E11:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E4:43"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="integra2.html#E4:43_2_1_1_2_2"><i><font color="Green" title="E46">A46</font></i></a>, <a class="txt" href="integra2.html#E6:43_2_1_1_2_2"><i><font color="Green" title="E47">A47</font></i></a></i>;<br/>
<a NAME="E12:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/>
<a NAME="E13:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E6:43_2_1_1_2_2"><i><font color="Green" title="E47">A47</font></i></a>, <a class="ref" href="goboard2.html#T18" title="GOBOARD2:th.18">GOBOARD2:18</a></i>;<br/>
<a NAME="E14:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E3:43_2_1_1_2_2"><i><font color="Green" title="E45">A45</font></i></a>, <a class="ref" href="goboard1.html#D1" title="GOBOARD1:def.1">GOBOARD1:def 1</a></i>;<br/>
<a NAME="E15:43_2_1_1_2_2"/><b>then </b><i><font color="Green" title="E48">A48</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E16:43_2_1_1_2_2"/>
<font color="Maroon" title="c8">i</font> <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#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E17:43_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="integra2.html#E15:43_2_1_1_2_2"><i><font color="Green" title="E48">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E18:43_2_1_1_2_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E13:43_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="integra2.html#E11:43_2_1_1_2"><i><font color="Green" title="E41">A41</font></i></a></i>;<br/><a NAME="E14:43_2_1_1_2"/><b>then </b><i><font color="Green" title="E43">A49</font></i>: 
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E15:43_2_1_1_2"/>
( <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E4:43"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="integra2.html#E8:43"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E10:43_2_1_1_2"><i><font color="Green" title="E40">A40</font></i></a>, <a class="txt" href="integra2.html#E12:43_2_1_1_2"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:43_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">i</font> ) </span>}</span> 
 ;<br/><a NAME="E17:43_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">D2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>hence </b><a NAME="E18:43_2_1_1_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="integra2.html#E3:43_2_1_1_2"><i><font color="Green" title="E36">A29</font></i></a>, <a class="txt" href="integra2.html#E6:43_2_1_1_2"><i><font color="Green" title="E38">A38</font></i></a>, <a class="txt" href="integra2.html#E14:43_2_1_1_2"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_2"/>
 <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c8">i</font>
;<br/>

<b>thus </b><a NAME="E14:43"/>
( <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="c2">D1</font> &amp;  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c3">D2</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="integra1.html#K2" title="INTEGRA1:func.2">divset</a> <font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="integra2.html#E11:43"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="integra2.html#E13:43"><i><font color="Green" title="E34">A11</font></i></a></i>;<br/>


</div>
