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<b>let </b><font color="Maroon" title="c1">E</font>, <font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>









<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/>

<b>given </b><font color="Maroon" title="c6">mn</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><a NAME="E3:52"/><i><font color="Green" title="E37">A2</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">mn</font> &amp; <font color="Maroon" title="c6">mn</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">mn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">mn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">mn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">mn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_1_1"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="flang_1.html#T28" title="FLANG_1:th.28">FLANG_1:28</a></i>;<br/><a NAME="E3:52_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="flang_2.html#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:52_1_1"/><i><font color="Green" title="E39">A4</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c7">i</font> )
 <b>by </b><i><a class="ref" href="flang_2.html#T19" target="_self" title="FLANG_2:th.19">Th19</a></i>;<br/><b>thus </b><a NAME="E6:52_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_2.html#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="flang_2.html#E2:52_1_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="flang_2.html#E5:52_1_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="flang_1.html#T28" title="FLANG_1:th.28">FLANG_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">mn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_1_2"/><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">mn</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 <b>by </b><i><a class="txt" href="flang_2.html#E3:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T41" title="ZFMISC_1:th.41">ZFMISC_1:41</a></i>;<br/><a NAME="E4:52_1_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="flang_2.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="flang_2.html#E3:52"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="flang_2.html#T19" target="_self" title="FLANG_2:th.19">Th19</a></i>;<br/><b>hence </b><a NAME="E5:52_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="flang_2.html#E1:52"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="flang_2.html#E3:52_1_2"><i><font color="Green" title="E38">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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