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<span class="kw">let </span><font color="Maroon" title="c3">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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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 <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">m</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">m</font> implies <font color="Maroon" title="c4">n</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">m</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:42"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">n</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">m</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font></span><br/>

<a NAME="E2:42"/><span class="kw">then </span>
( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> or <font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E3:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 <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="c3">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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="c3">L</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sfmastr1.html#T19" target="_self" title="SFMASTR1:th.19">Th20</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:42"/>
<font color="Maroon" title="c4">n</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="sfmastr1.html#K6" title="SFMASTR1:func.6">-thRWNotIn</a> <font color="Maroon" title="c3">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">hence </span><a NAME="E5:42"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="ref" href="ami_3.html#T10" title="AMI_3:th.10">AMI_3:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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