<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<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#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:53"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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>
 <span class="kw">and </span><br/><a NAME="E4:53"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_2.html#K4" title="AFINSQ_2:func.4">Sgm0</a> <font color="Maroon" title="c5">X</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_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>
 <span class="kw">and </span><br/><a NAME="E5:53"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:53"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:53"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="afinsq_2.html#D4" target="_self" title="AFINSQ_2:def.4">Def5</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/>


</div>
