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<span class="kw">let </span><font color="Maroon" title="c1">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not  <a href="sf_mastr.html#K5" title="SF_MASTR:func.5">FirstNotIn</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">FNI</font> =  <a href="sf_mastr.html#K5" title="SF_MASTR:func.5">FirstNotIn</a> <font color="Maroon" title="c1">L</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">sn</font> being  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> <span class="kw"> such that </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 <a href="sf_mastr.html#K5" title="SF_MASTR:func.5">FirstNotIn</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <span class="p1">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">sn</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:65"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c3">sn</font> <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="c1">L</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sf_mastr.html#D6" target="_self" title="SF_MASTR:def.6">Def6</a></span>;<br/>
<a NAME="E4:65"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">sn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">sn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" title="XXREAL_2:def.7">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E5:65"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">sn</font> &amp; 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="c1">L</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:65"/>
not  <a href="sf_mastr.html#K5" title="SF_MASTR:func.5">FirstNotIn</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E46">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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