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<b>let </b><font color="Maroon" title="c1">E</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">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="c3">m</font>, <font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>thus </b><a NAME="E1:22"/>
(  not <font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</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">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> or ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:22_1"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:22_1"/><i><font color="Green" title="E15">A2</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 or <font color="Maroon" title="c2">A</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><i><font color="Green" title="E16">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><a NAME="E2:22_1_1"/><b>then </b>
<span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><span class="p2">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="flang_1.html#T30" title="FLANG_1:th.30">FLANG_1:30</a></i>;<br/><a NAME="E3:22_1_1"/><b>then </b>
 <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">m</font>
 <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="E4:22_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_2.html#E1:22_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="flang_2.html#E1:22_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="flang_2.html#T19" target="_self" title="FLANG_2:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_2"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c2">A</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/><a NAME="E2:22_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">m</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> 
 <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="E3:22_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">a</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>hence </b><a NAME="E4:22_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_2.html#E1:22_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="flang_2.html#E1:22_1_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="flang_2.html#T19" target="_self" title="FLANG_2:th.19">Th19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:22_1"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_2.html#E2:22_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="flang_2.html#E3:22_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<div><i><font color="Green" title="E14">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:22_2_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> or  not <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="flang_2.html#E1:22_2"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E2:22_2_1"/>
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="flang_2.html#T19" target="_self" title="FLANG_2:th.19">Th19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:22_2"/>
<font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</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="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_3"/><i><font color="Green" title="E15">A8</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">A</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:22_3_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:22_3_1"/><i><font color="Green" title="E16">A9</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c6">k</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="E4:22_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="flang_2.html#E1:22_3"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="flang_2.html#E3:22_3_1"><i><font color="Green" title="E16">A9</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><b>hence </b><a NAME="E3:22_3"/>
<font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</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="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
( ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">n</font> or ( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) ) implies <font color="Maroon" title="c2">A</font> <a href="flang_2.html#K1" title="FLANG_2:func.1">|^</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</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="txt" href="flang_2.html#E2:22"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>


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