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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>assume </b><a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
(  <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> &amp; <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> )
 ;<br/>

<a NAME="E2:35"/><i><font color="Green" title="E26">A2</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E27">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1_1"/>
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> )
 </span><b>by </b><i><a class="txt" href="flang_2.html#E1:35_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c5">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="flang_2.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="flang_1.html#T37" title="FLANG_1:th.37">FLANG_1:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_1_2"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c5">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:35_1"/><i><font color="Green" title="E28">A5</font></i>: 
<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>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="E4:35_1"/><i><font color="Green" title="E29">A6</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="txt" href="flang_2.html#E2:35_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="flang_2.html#T19" target="_self" title="FLANG_2:th.19">Th19</a></i>;<br/>
<a NAME="E5:35_1"/>
<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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="flang_2.html#E4:35_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="flang_2.html#E1:35_1"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:35_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_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="flang_2.html#E4:35_1"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:35"/>
<font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="txt" href="flang_2.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="flang_2.html#T20" target="_self" title="FLANG_2:th.20">Th20</a></i>;<br/>
<b>hence </b><a NAME="E4:35"/>
<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> <font color="Maroon" title="c2">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="flang_2.html#E2:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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