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

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)</span>)</span></span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b1">p</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> );<br/>
<span class="kw">set </span><font color="Maroon" title="c3">F</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Z</font> =  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>);<br/>
<a NAME="E1:49"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">pN</font> where <font color="Olive" title="b2">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b2">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b3">N</font> where <font color="Olive" title="b3">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b3">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b3">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:49_1"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T33" title="AFINSQ_1:th.33">AFINSQ_1:33</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<a NAME="E2:49_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T53" title="AFINSQ_2:th.53">AFINSQ_2:53</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:49_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">pN</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E5:49_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pN</font> &amp; <font color="Maroon" title="c6">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b1">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>
<a NAME="E6:49_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">pN</font>
 
;<br/>
<a NAME="E7:49_1"/>
( <font color="Maroon" title="c6">pN</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">q</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <span class="kw"> such that </span><br/><a NAME="E9:49_1"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c6">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">and </span><br/><a NAME="E10:49_1"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c7">q</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E6:49_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="catalan2.html#T17" target="_self" title="CATALAN2:th.17">Th21</a></span>;<br/>
<a NAME="E11:49_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>;<br/>
<a NAME="E12:49_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A8</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E1:49_1"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">q</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">q</font>] )</span><br/>

<a NAME="E13:49_1"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c6">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:49_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/>
<a NAME="E14:49_1"/><span class="kw">then </span>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E2:49_1"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:49_1"/>
( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">q</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E9:49_1"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E10:49_1"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E12:49_1"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> ,<span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:49"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">pN</font> where <font color="Olive" title="b2">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b2">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b3">N</font> where <font color="Olive" title="b3">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b3">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b3">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E41">A1</font></span></a>);<br/></span>
<a NAME="E4:49"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
(  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <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="49_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:49_2"/>
 <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:49_2"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#T22" target="_self" title="CATALAN2:th.22">Th26</a></span>;<br/>
<a NAME="E3:49_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A11</font></span>: 
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:49_2"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:49_2"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">pN</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E8:49_2"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pN</font> &amp; <font color="Maroon" title="c7">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b1">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_2"><span class="lab"><font color="Green" title="E45">A12</font></span></a></span>;<br/>
<a NAME="E9:49_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">pN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<a NAME="E10:49_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T69" title="RELAT_1:th.69">RELAT_1:69</a></span>;<br/>
<a NAME="E11:49_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<a NAME="E12:49_2"/>
<font color="Maroon" title="c7">pN</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<a NAME="E13:49_2"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:49_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="ref" href="catalan2.html#T2" target="_self" title="CATALAN2:th.2">Th6</a></span>;<br/>
<a NAME="E14:49_2"/><span class="kw">then </span>
2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:49_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:49_2"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E15:49_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b1">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> } </span> 
 
;<br/>
<span class="kw">hence </span><a NAME="E16:49_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:49"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:49_3"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:49_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <span class="kw"> such that </span><br/><a NAME="E4:49_3"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E5:49_3"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c7">p</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> 
 <span class="kw">and </span><br/><a NAME="E6:49_3"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E7:49_3"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalan2.html#D2" target="_self" title="CATALAN2:def.2">Def2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">q</font> = <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font>;<br/>
<a NAME="E8:49_3"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N</font> &amp; <font color="Olive" title="b1">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_3"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="catalan2.html#T18" target="_self" title="CATALAN2:th.18">Th22</a></span>;<br/>
<a NAME="E9:49_3"/>
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c7">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T55" title="AFINSQ_2:th.55">AFINSQ_2:55</a></span>;<br/>
<a NAME="E10:49_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A22</font></span>: 
 <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:49_3"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="afinsq_2.html#T53" title="AFINSQ_2:th.53">AFINSQ_2:53</a></span>;<br/>
<a NAME="E11:49_3"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" title="AFINSQ_1:def.7">AFINSQ_1:def 7</a></span>;<br/>
<a NAME="E12:49_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c7">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D3" title="AFINSQ_1:def.3">AFINSQ_1:def 3</a></span>;<br/>
<a NAME="E13:49_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A24</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:49_3"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="afinsq_1.html#T33" title="AFINSQ_1:th.33">AFINSQ_1:33</a></span>;<br/>
<a NAME="E14:49_3"/>
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font> is  <a href="catalan2.html#V1" title="CATALAN2:attr.1">dominated_by_0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49_3"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="catalan2.html#T18" target="_self" title="CATALAN2:th.18">Th22</a></span>;<br/>
<a NAME="E15:49_3"/><span class="kw">then </span>
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:49_3"><span class="lab"><font color="Green" title="E53">A24</font></span></a>, <a class="txt" href="#E10:49_3"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="catalan2.html#T20" target="_self" title="CATALAN2:th.20">Th24</a></span>;<br/>
<a NAME="E16:49_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:49_3"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E11:49_3"><span class="lab"><font color="Green" title="E52">A23</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <span class="kw"> such that </span><br/><a NAME="E18:49_3"/><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">and </span><br/><a NAME="E19:49_3"/><span class="lab"><font color="Green" title="E56">A27</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>;<br/>
<a NAME="E20:49_3"/>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:49_3"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="ref" href="afinsq_1.html#T28" title="AFINSQ_1:th.28">AFINSQ_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:49_3"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E4:49_3"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E16:49_3"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="txt" href="#E19:49_3"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:49"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">pN</font> where <font color="Olive" title="b3">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b3">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b4">N</font> where <font color="Olive" title="b4">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b4">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b4">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">pN</font> where <font color="Olive" title="b3">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b3">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b4">N</font> where <font color="Olive" title="b4">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b4">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">N</font> &amp; <font color="Olive" title="b4">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:49_4"/><span class="lab"><font color="Green" title="E46">A28</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  )
 <span class="kw">and </span><br/><a NAME="E2:49_4"/><span class="lab"><font color="Green" title="E47">A29</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/>

<a NAME="E3:49_4"/>
(  ex <font color="Olive" title="b1">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> ) &amp;  ex <font color="Olive" title="b1">q</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  st <br/>( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Olive" title="b1">q</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E1:49_4"><span class="lab"><font color="Green" title="E46">A28</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:49_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:49_4"><span class="lab"><font color="Green" title="E47">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:49"/><span class="kw">then </span>
<font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a></span>;<br/>
<a NAME="E9:49"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span> , <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E6:49"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:49"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pN</font> where <font color="Olive" title="b1">pN</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  : ( <font color="Olive" title="b1">pN</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Maroon" title="c2">m</font>) &amp; <span class="p1"> { <span class="default"> <font color="Olive" title="b2">N</font> where <font color="Olive" title="b2">N</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 2 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">pN</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">N</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N</font> &amp; <font color="Olive" title="b2">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> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="catalan2.html#K1" title="CATALAN2:func.1">Domin_0</a> (<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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