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<b>let </b><font color="Maroon" title="c1">l</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">X</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> holds <br/><font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font></span><br/><b>let </b><font color="Maroon" title="c5">X</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="membered.html#V6" title="MEMBERED:attr.6">natural-membered</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">l</font> &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> implies <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green" title="E41">A2</font></i>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">l</font>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span>
 <b>and </b><br/><a NAME="E3:59"/><i><font color="Green" title="E43">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K9" title="AFINSQ_2:func.9">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l</font> )
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green" title="E44">A6</font></i>: 
not <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:59"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="#E3:59"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="#E4:59"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E6:59"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:59"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="#E2:59"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E3:59"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="afinsq_2.html#D5" target="_self" title="AFINSQ_2:def.5">AC113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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