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<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  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">sn</font> = <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> ;<br/>
<a NAME="E1:64_1_1"/><i><font color="Green">E32</font></i>: 
<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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E2:64_1_1"/>
not  <a href="scmfsa_2.html#K2">Int-Locations</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#T22">SCMFSA_2:22</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:64_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K2">Int-Locations</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="scmfsa_2.html#M1">Int-Location</a>  <b>by </b><i>, <a class="ref" href="scmfsa_2.html#T11">SCMFSA_2:11</a></i>;<br/>
<b>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="E7:64_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T19">SCMFSA_2:19</a></i>;<br/>
<a NAME="E8:64_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">sn</font> = <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>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] </span>}</span>  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>  <b>by </b><i/>;<br/>
<b>take </b>
 <a href="scmfsa_2.html#K4">intloc</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">sn</font></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">sn</font>
;<br/>

<b>thus </b><a NAME="E10:64_1_1"/>
(  <a href="scmfsa_2.html#K4">intloc</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">sn</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K4">intloc</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">sn</font></span>)</span> &amp; <font color="Maroon">sn</font> <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> </span>}</span>  )
 ;<br/>


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