<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>   holds  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive" title="b1">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><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>set </b><font color="Maroon" title="c5">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="c3">n</font>;<br/>
<b>set </b><font color="Maroon" title="c6">sn1</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E1:39"/><i><font color="Green" title="E21">A1</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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4: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/>
<a NAME="E2:39"/><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <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;  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green" title="E23">A3</font></i>: 
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="seq_4.html#T64" title="SEQ_4:th.64">SEQ_4:64</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>assume </b><a NAME="E4:39"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:39"/><b>then </b>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sfmastr1.html#E3:39"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<a NAME="E6:39"/><b>then </b>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p3">(<span class="default"><span class="p4">(<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></span>)</span></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:39"/>
contradiction
 <b>by </b><i><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">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
