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<b>let </b><font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> );<br/><a NAME="E1:41_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E2:41_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> implies  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E2:41_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:41_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> )
 </span><b>by </b><i><a class="txt" href="sfmastr1.html#E3:41_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1_1"/>
 <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sfmastr1.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_2"/><b>then </b>
(  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> &amp;  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="sfmastr1.html#E1:41_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="sfmastr1.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_1_2"/>
 <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:41_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="sfmastr1.html#E1:41_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="sfmastr1.html#E2:41_1"><i><font color="Green">E24</font></i></a>);</i><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> implies  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sfmastr1.html#K5" target="_self">RWNotIn-seq</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/>


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