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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">E</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="c2">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>thus </b><a NAME="E1:54"/>
( <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <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> implies ( <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; ( (  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E38">A1</font></i>: 
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <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>
 ;<br/>

<b>then consider </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><br/><a NAME="E3:54_1"/><i><font color="Green" title="E39">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; <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <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> )
 <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="E4:54_1"/>
( <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; ( (  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> ) ) )
 <b>by </b><i><a class="txt" href="flang_2.html#E1:54_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="flang_2.html#E3:54_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="flang_2.html#T9" target="_self" title="FLANG_2:th.9">Th9</a>, <a class="ref" href="flang_2.html#T21" target="_self" title="FLANG_2:th.21">Th21</a></i>;<br/>


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

<b>assume </b><a NAME="E2:54"/><i><font color="Green" title="E38">A3</font></i>: 
( <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; ( (  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> ) ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_2"/>
( (  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> ) )
 </span><b>by </b><i><a class="txt" href="flang_2.html#E2:54"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1"><b>suppose </b></a><a NAME="E1:54_2_1"/>
(  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c2">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><div class="add"><a NAME="E2:54_2_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="ref" href="flang_1.html#T36" title="FLANG_1:th.36">FLANG_1:36</a></i>;<br/><b>hence </b><a NAME="E3:54_2_1"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <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#E2:54"><i><font color="Green" title="E38">A3</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="hsNdiv(this)" href="javascript:()" title="54_2_2"><b>suppose </b></a><a NAME="E1:54_2_2"/><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> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> )
 ;<br/><div class="add"><a NAME="E2:54_2_2"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <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> 1
 <b>by </b><i><a class="txt" href="flang_2.html#E2:54"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="flang_1.html#T26" title="FLANG_1:th.26">FLANG_1:26</a></i>;<br/><b>hence </b><a NAME="E3:54_2_2"/>
<span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span> <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#E1:54_2_2"><i><font color="Green" title="E39">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><b>end;</b></div>

</div>
