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<b>let </b><font color="Maroon">L</font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="scmfsa_2.html#K2">Int-Locations</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">RL</font> =  <a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( not 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> ) );<br/>
<a NAME="E1:39"/><i><font color="Green">E35</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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> 0
 ;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 &amp; not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:39_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:39_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:39_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E6:39_1"/>
not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:39"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_2"/><i><font color="Green">E40</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:39_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> holds <br/> not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">sn</font> = <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E4:39_2"/><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">sn</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">sn</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:39_2"/>
not 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E6:39_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E7:39_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:39_2"/>
not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="sfmastr1.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:39"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:39"/>
( not 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> holds <br/> not  <a href="scmfsa_2.html#K4">intloc</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> ) )
 ;<br/>


</div>
