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

<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">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> );<br/>
<a NAME="E1:20"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E18">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>let </b><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="E1:20_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1"/>
( <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>thus </b><a NAME="E2:20_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="flang_3.html#E1:20_1_1_1_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="flang_1.html#T26" title="FLANG_1:th.26">FLANG_1:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
<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"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c4">m</font></span>)</span>
<b>by </b><i><a class="txt" href="flang_3.html#E1:20_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="flang_3.html#E1:20_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="flang_1.html#T24" title="FLANG_1:th.24">FLANG_1:24</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">m</font></span>)</span>
<b>by </b><i><a class="ref" href="flang_3.html#T18" target="_self" title="FLANG_3:th.18">Th18</a></i>
<br/>.= 
<font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>

;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:20"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="flang_3.html#E1:20"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="flang_3.html#E2:20"><i><font color="Green" title="E18">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:20"/>
( <font color="Maroon" title="c3">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>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="flang_1.html#K7" title="FLANG_1:func.7">|^</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="flang_3.html#K1" title="FLANG_3:func.1">|^..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">n</font></span>)</span> )
 ;<br/>


</div>
