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

<b>let </b><font color="Maroon" title="c2">L</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="scmfsa_2.html#K2" title="SCMFSA_2:func.2">Int-Locations</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c4">RL</font> =  <a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 holds <br/> not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> ) );<br/>
<a NAME="E1:39"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<a NAME="E1:39_1"/><i><font color="Green" title="E22">A2</font></i>: 
<span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="sfmastr1.html#D2" target="_self" title="SFMASTR1:def.2">Def2</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1_1"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0
 ;<br/><b>then consider </b><font color="Maroon" title="c5">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c5">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="sfmastr1.html#E1:39_1"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E4:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="sfmastr1.html#E3:39_1_1"><i><font color="Green" title="E23">A3</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">m</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="E3:39_1"/>
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">k</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="E5:39_1"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> &amp; not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="sfmastr1.html#E1:39_1"><i><font color="Green" title="E22">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:39_1"/>
not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="sfmastr1.html#E5:39_1"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:39"/><i><font color="Green" title="E22">A5</font></i>: 
for <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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:39_2"/><i><font color="Green" title="E23">A6</font></i>: 
not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 <b>and </b><br/><a NAME="E2:39_2"/><i><font color="Green" title="E24">A7</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font> holds <br/> not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">sn</font> = <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</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="E4:39_2"/><i><font color="Green" title="E25">A8</font></i>: 
<span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">sn</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c6">sn</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="sfmastr1.html#D2" target="_self" title="SFMASTR1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:39_2"/>
not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="sfmastr1.html#E1:39_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">m</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="E6:39_2"/>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E7:39_2"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">n</font>
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E4:39_2"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:39_2"/>
not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="sfmastr1.html#E2:39_2"><i><font color="Green" title="E24">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:39"/>
for <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="sfmastr1.html#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="sfmastr1.html#E2:39"><i><font color="Green" title="E22">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:39"/>
( not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> &amp; ( for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" title="SFMASTR1:func.5">RWNotIn-seq</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> holds <br/> not  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> ) )
 ;<br/>


</div>
